SPEER'S 

ARITHMETICS 


PRIMARY... 

f  OR  TEACHERS 


GIMN  AND  COMPANY 


ji—h-^TL., 


"L-fi—n — n, 


REESE  LIBRARY  j 

OF  THK 

UNIVERSITY  OF  CALIFORNIA. 


Vr—  v—  u 


Deceive  J 


Accessions  No. 


.     OIKS  No. 


PRIMAKY 

AKITHMETIC 


FIRST  YEAR 


FOB   THE   USE   OF   TEACHERS 


BY 

WILLIAM   W.  SPEEK 

ASSISTANT  SUPERINTENDENT  OF  SCHOOLS,  CHICAGO 


BOSTON,  U.S.A.,  AND  LONDON 
GINN  &   COMPANY,  PUBLISHERS 


1897 


L  (6  1534- 

S7 


COPYRIGHT,   1896 

BY  WILLIAM  W.  SPEER 


ALL  RIGHTS   RESERVED 


PREFACE. 


THIS  book  is  one  of  a  series  soon  to  be  issued.  The 
point  of  view  from  which  it  is  written  is  indicated  in 
the  introduction. 

The  essence  of  the  theory  of  teaching  arithmetic  can 
be  expressed  in  a  few  sentences.  The  fundamental  thing 
is  to  induce  judgments  of  relative  magnitude.  The  pres- 
entation regards  the  fact  that  it  is  the  relation  of  things 
that  makes  them  what  they  are.  The  one  of  mathematics 
is  not  an  individual,  separated  from  all  else,  but  the  union 
of  two  like  impressions  :  the  relation  of  two  equal  magni- 
tudes. A  child  does  not  perceive  this  one  until  he  sees  the 
equality  of  two  magnitudes.  He  will  not  become  sensitive 
to  relations  of  equality  by  handling  equal  units  with  the 
attention  directed  to  something  else,  as  the  color,  the  tex- 
ture, or  the  how  many  ;  nor  by  one  or  two  experiences  in 
comparing  magnitudes. 

To  aid  the  learner  in  seeing  a  1  as  the  relation  of  two 
equal  units,  a  2  as  the  relation  of  a  unit  to  another  one  half 
as  large,  a  %  as  the  relation  of  a  unit  to  another  twice  as 
large,  we  must  induce  the  repeated  acts  of  comparing  which 
bring  these  relations  vividly  before  the  mind.  With  this 
purpose  the  child  is  not  required  to  build  out  of  parts  a 
whole  which  he  has  never  seen,  nor  expected  to  discover 
a  relation  in  the  absence  of  one  of  the  related  terms.  He 
does  not  begin  with  elements.  He  is  not  prevented  from 


IV  PREFACE. 

seeing  things  as  they  are  by  pushing  elements  into  the 
foreground.  The  mind  grasps  something  vaguely  as  a 
whole,  moves  from  this  to  the  parts,  and  gradually  advances 
to  a  clearer  and  fuller  idea  of  the  whole.  Whether  the 
object  of  study  be  a  flower,  a  picture,  a  cubic  foot,  or  a  six, 
the  process  of  learning  is  the  same.  If  we  promote  prog- 
ress in  the  discovery  of  relations  of  magnitude,  we  will 
make  it  possible  for  the  compared  wholes  to  be  pictured  in 
their  full  extent,  thus  affording  opportunity  for  comparing, 
for  activity  in  judging.  There  is  no  such  opportunity 
when  a  child  who  has  no  idea  of  a  thing  constructs  it 
mechanically  from  given  parts.  Creation,  in  any  subject, 
requires  a  basis  in  elementary  ideas. 

It  is  not  to  be  forgotten  that  there  is  a  wide  difference 
between  seeing  that  the  relation  of  two  particular  things  is 
8,  and  realizing  8  as  a  relation,  realizing  it  in  such  a  way 
that  it  can  be  freely  used  without  misapplying  it. 

There  is  no  real  progress  unless  the  mind  is  gradually 
gaining  power  to  think  of  things  not  present  to  sense,  and 
to  think  of  a  relation  apart  from  a  particular  thing.  But 
there  is  no  way  to  promote  this  progress  except  by  securing 
continued  activity  of  sense  and  mind.  The  child  grows 
into  the  idea  8,  slowly  and  unconsciously  but  surely,  under 
right  conditions.  A  cube  does  not  become  known  by  count- 
ing surfaces,  edges,  etc.,  again  and  again,  but  by  observing 
other  forms  and  many  different  cubes.  Through  repeated 
acts  of  dissociating  and  relating,  what  is  particular  sinks 
out  of  sight  and  the  common  trait  stands  out.  This 
principle  is  of  general  application. 

There   should   be  constant   calls  for   reperception,  for 

judging  and  verifying.     Only  by  multiplying  experiences 

in  the  concrete,  by  noting  the  same  relation  in  many  dif- 

\  ferent  things  and  in  many  different  conditions,  does  the 

child  come  to  know  a  relation  as  it  is. 


PREFACE.  Y 

The  slow  development  of  the  power  to  form  perfectly 
quantitative  judgments  is  considered.  Hence  the  earlier 
work  makes  no  demand  for  close  analysis.  It  provides  for 
a  gradual  advance  toward  exactness.  The  exercises  are 
only  suggestive.  The  condition  of  the  child  determines 
what  he  should  do.  But  in  any  case,  the  work  in  the 
beginning  should  be  so  simple  that  it  can  be  done  easily  ; 
it  should  look  to  the  free  action  of  both  body  and  mind. 

The  child  interested  in  finding  colors  and  forms  wishes 
to  move  about,  to  touch  and  handle  things.  Out  of  school 
he  combines  thinking  and  acting.  Why  should  he  not  do 
so  in  the  school  ?  Interest  will  lead  the  child  to  control 
himself,  but  repression  from  without  induces  dullness, 
indifference,  and  antagonism.  Force  a  child  to  preserve  a 
regulation  attitude,  to  keep  his  nerves  tense,  and  you  destroy 
the  foundation  of  healthful  mental  activity.  In  the  transi- 
tion from  home  to  school  life,  careful  provision  should  be 
made  for  the  whole  child  to  express  himself. 

Attention  is  asked  to  the  remarks  upon  over-direction, 
premature  questioning,  demands  for  analysis  beyond  the 
inclination  and  power  of  the  pupil  and  for  outward  forms 
which  are  not  the  genuine  expression  of  the  child. 

Great  importance  is  attached  to  that  order  of  work  which 
puts  things  before  the  pupil  and  leaves  him  free  to  see  and 
to  tell  all  he  can  before  interfering  with  his  action  by  ques- 
tioning or  direction.  Questions  have  their  uses.  They 
serve  to  arouse  attention,  to  aid  in  testing  the  pupil's  view, 
and  may  lead  to  the  correct  use  of  new  forms  of  expression. 
But  there  are  effects  of  questioning  which  are  too  often 
overlooked.  Questions  do  for  the  child  what  he  should  do 
for  himself ;  they  conceal  his  attitude  toward  the  work 
and  prevent  your  seeing  what  he  would  do  unaided.  They 
call  attention  to  details  for  which  the  mind  may  not  be 
prepared,  and  present  a  partial,  fragmentary  view.  The 


VI  PREFACE. 

questioning  may  be  logical,  but  the  learner  connects  only 
that  which  he  himself  relates.  Questions  cause  the  teacher 
to  suppose  that  the  child  grasps  what  is  not  appreciable  by 
him,  and  so  prevent  the  adaptation  of  the  work.  To 
attempt  to  force  through  questions  what  you  see  in  a 
poem,  picture,  or  problem,  instead  of  leaving  the  pupil  to 
discover  what  he  is  prepared  to  see,  is  to  ignore  the  true 
basis  of  advance,  to  disregard  the  law  that  the  mind  passes 
from  vague  ideas  to  those  fuller  and  more  exact,  only 
through  its  own  acts  of  analysis  and  synthesis.  Free 
work  reveals  the  pupil  and  makes  it  possible  to  meet  his 
needs. 

This  view  furnishes  no  excuse  for  random,  desultory 
work.  The  teacher  must  carefully  select  the  means,  whether 
the  ideas  into  which  he  wishes  to  lead  the  child  are  mathe- 
matical, biological,  or  historical. 

In  conclusion  it  is  urged  that  any  success  is  dangerous 
which  lessens  the  susceptibility  of  the  mind  to  new  impres- 
sions. We  may  be  so  successful  in  training  the  child  to 
reproduce  as  to  destroy  his  power  to  produce.  Progress  is 
impossible  without  growing  power  to  do  unconsciously  what 
was  at  first  done  consciously  ;  but  accuracy  is  not  to  be 
desired  at  the  expense  of  growth.  The  purpose  of  auto- 
matic action  in  education  is  not  to  restrict,  but  to  set  force 
free.  When  the  work  of  the  school  is  mechanical  it 
weakens  the  relating  power,  the  power  to  act  in  new  cir- 
cumstances, and  thus  lowers  the  child  in  the  scale  of 
being. 

As  insight  into  the  subject  and  contact  with  the  child 
enable  us  to  open  right  channels  for  free  action,  there 
will  be  little  occasion  for  drills.  The  fresh,  vigorous  effort 
of  involuntary  attention  carries  the  child  forward  with  sur- 
prising rapidity.  Out  of  se(/*-activity  comes  the  self-control 
which  gives  strength  to  persist. 


THEOET    OF    AEITHMETIO. 


DSTTKODUCTIOK 

THE  following  quotations  may  be  found  sugges- 
tive of  working  ideals.  The  teacher  who  enters 
into  their  spirit  will  feel  the  need  of  knowing  both 
the  child  and  the  subject.  She  will  see  that 
attention  is  a  condition  of  thinking,  and  interest 
a  condition  of  attention  ;  that  the  mind  is  one 
and  indivisible,  and  must  be  so  treated  if  we  would 
strengthen  it.  The  mental  as  well  as  the  physical 
nature  is  under  law.  When  our  teaching  is  in 
accord  with  this  law,  we  shall  find  the  forces  of 
nature  working  for  us ;  the  child  will  become 
strong  with  the  strength  of  nature. 

Apprehension  by  the  senses  supplies,  directly  or 
indirectly,  the  material  of  all  human  knowledge  ;  or, 
at  least,  the  stimulus  necessary  to  develop  every  inborn 
faculty  of  the  mind.  —  Helmholtz. 

The  products  of  the  senses,  especially  those  of  sight, 
hearing,  and  touch,  form  the  basis  of  all  the  higher 
thought  processes.  Hence  the  importance  of  developing 
accurate  sense  concepts.  .  .  .  The  purpose  of  objective 


2  THEORY    OF    ARITHMETIC. 

thinking  is  to  enable  the  mind  to  think  without  the  help 
of  objects.  —  Thomas  M.  Balliet. 

The  understanding  must  begin  by  saturating  itself 
with  facts  and  realities.  .  .  .  Besides,  we  only  under- 
stand that  which  is  already  within  us.  To  understand 
is  to  possess  the  thing  understood,  first  by  sympathy 
and  then  by  intelligence.  Instead  of  first  dismember- 
ing and  dissecting  the  object  to  be  conceived,  we  should 
begin  by  laying  hold  of  it  in  its  ensemble.  The  pro- 
cedure is  the  same,  whether  we  study  a  watch  or  a 
plant,  a  work  of  art  or  a  character.  — Amiel. 

The  action  of  the  mind  in  the  acquisition  of  knowl- 
edge of  any  sort  is  synthetic-analytic ;  that  is,  uniting 
and  separating.  These  are  the  two  sides,  or  aspects,  of 
the  one  process.  .  .  .  There  is  no  such  thing  as  a  syn- 
thetic activity  that  is  not  accompanied  by  the  analytic  ; 
and  there  is  no  analytic  activity  that  is  not  accompanied 
by  the  synthetic.  Children  cannot  be  taught  to  perform 
these  knowing  acts.  It  is  the  nature  of  the  mind  to  so 
act  when  it  acts  at  all.  —  George  P.  Brown. 

Our  children  will  attain  to  a  far  more  fundamental 
insight  into  language,  if  we,  when  teaching  them,  con- 
nect the  words  more  with  the  actual  perception  of  the 
thing  and  the  object.  .  .  .  Our  language  would  then 
again  become  a  true  language  of  life,  that  is,  born  of 
life  and  producing  life.  —  FroebeL 

Voluntary  attention  is  a  habit,  an  imitation  of  natural 
attention,  which  is  its  starting-point  and  its  basis.  .  .  . 


INTRODUCTION.  3 

Attention  creates  nothing  ;  and  if  the  brain  is  barren, 
if  the  associations  are  meagre,  it  functions  in  vain.  — 
Eibot. 

How,  indeed,  can  there  be  a  response  within  to  the 
impression  from  without  when  there  is  nothing  within 
that  is  in  relation  of  congenial  vibration  with  that 
which  is  without?  Inattention  in  such  case  is  insus- 
ceptibility ;  and  if  this  be  complete,  then  to  demand 
attention  is  very  much  like  demanding  of  the  eye  that 
it  should  attend  to  sound-waves,  and  of  the  ear  that  it 
should  attend  to  light-waves.  —  Dr.  Maudsley. 

Activity  bears  fruit  in  habit,  and  the  kind  of  activity 
determines  the  quality  of  the  habit.  —  Alex  JEJ.  Frye. 

If  a  teacher  is  full  of  his  subject,  and  can  induce 
enthusiasm  in  his  pupils ;  if  his  facts  are  concrete  and 
naturally  connected,  the  amount  of  material  that  an 
average  child  can  assimilate  without  injury  is  as  aston- 
ishing as  is  the  little  that  will  fag  him  if  it  is  a  trifle 
above  or  below  or  remote  from  him,  or  taught  dully  or 
incoherently.  —  6r.  Stanley  Hall. 

Is  it  not  evident,  that  if  the  child  is  at  any  epoch  of 
his  long  period  of  helplessness  inured  into  any  habit 
or  fixed  form  of  activity  belonging  to  a  lower  stage  of 
development,  the  tendency  will  be  to  arrest  growth 
at  that  standpoint  and  make  it  difficult  or  next  to 
impossible  to  continue  the  growth  of  the  child  ?  —  Wil- 
liam T.  Harris. 


4  THEORY    OF    ARITHMETIC. 

We  must  make  practice  in  thinking,  or,  in  other 
words,  the  strengthening  of  reasoning  power,  the 
constant  object  of  all  teaching  from  infancy  to 
adult  age,  no  matter  what  may  be  the  subject  of 
instruction.  .  .  .  Effective  training  of  the  reasoning 
powers  cannot  be  secured  simply  by  choosing  this 
subject  or  that  for  study.  The  method  of  study  and 
the  aim  in  studying  are  the  all-important  things.— 
Charles  W.  Eliot. 

Intellectual  evolution  is,  under  all  its  aspects,  a 
progress  in  representativeness  of  thought.  —  Herbert 
Spencer. 

Consciousness  implies  perpetual  discrimination,  or 
the  recognition  of  likenesses  and  differences,  and  this 
is  impossible  unless  impressions  persist  long  enough  to 
be  compared  with  one  another.  .  .  .  Impressions  persist 
long  enough  to  be  compared  together,  and  accordingly 
there  is  reason.  —  John  Fiske. 

Thinking  is  discerning  relations  ;  but  we  discern  the 
relations  of  things.  In  order  to  discern  relations  we 
must  compare  ;  hence,  our  powers  to  think  are  our 
comparative  powers.  These  are  our  faculties  to  discern 
relations.  —  Dr.  M' Cosh. 

Thought  consists  in  the  establishment  of  relations. 
There  can  be  no  relation  established,  and  therefore  no 
thought  framed  when  one  of  the  related  terms  is  absent 
from  consciousness.  —  Herbert  Spencer. 


INTRODUCTION.  5 

Intelligence  is  virtually   a   correct   classification.— 
Dr.  Maudsley. 

The  thing  is  its  relations,  and  although  analytically 
we  may  separate  them,  attending  now  to  this  relation, 
now  to  that,  we  must  never  imagine  the  separation  to 
be  real.  —  Gr.  H.  Lewes. 

All  knowledge  results  from  the  establishment  of 
relations  between  phenomena.  —  J.  B.  Stallo. 

Every  act  of  judgment  is  an  attempt  to  reduce  to 
unity  two  cognitions.  —  Sir  William  Thomson. 

The  primary  element  of  all  thought  is  a  judgment 
which  arises  from  a  comparison.  —  Francis  Bow  en. 

There  is  no  enlargement  of  the  mind  unless  there 
be  a  comparison  of  ideas  one  with  another.  —  Cardinal 
Newman. 

The  extent  or  magnitude  of  a  quantity  is,  therefore, 
purely  relative,  and  hence  we  can  form  no  idea  of  it 
except  by  the  aid  of  comparison.  —  Davies. 

Of  absolute  magnitude  we  can  frame  no  conception. 
All  magnitudes  as  known  to  us  are  thought  of  as  equal 
to,  greater  than,  or  less  than,  certain  other  magnitudes. 
—  Herbert  Spencer. 

Those  who  accept  the  above  can  hardly  agree 
with  the  prevailing  practices  in  the  teaching  of 
arithmetic. 


6  THEORY    OF    ARITHMETIC. 

It  is  hoped  that  the  following  brief  presenta- 
tion of  mathematics  as  the  science  of  relative 
magnitude  will  aid  teachers  in  bringing  mathe- 
matical teaching  into  accord  with  educational 
principles. 


MATHEMATICS,  —  DEFINITE   KELATIONS. 

Mental  advance  from  the  vague  to  the  definite.  - 

Teaching  which  meets  the  needs  of  the  developing 
mind  must  be  successful.  No  other  can  be. 

The  marvelous  progress  of  the  child  during  the 
first  five  or  six  years  of  its  life  is  largely  due  to 
free  action  and  spontaneous  attention ;  to  the 
absence  of  demands  unfavorable  to  growth. 

We  recognize  the  incapacity  of  the  infant.  We 
watch  and  minister  to  its  growth  by  creating  an 
environment  fitted  for  calling  forth  its  activities. 
So  should  we  acquaint  ourselves  with  the  mental 
state  of  the  child,  as  shown  in  his  work,  his  play, 
his  questions ;  in  what  he  hears  and  sees ;  in  what 
he  does  and  in  what  he  tries  to  do ;  in  what  he 
says  and  in  what  he  does  not  say.  From  the 
basis  of  his  experience  and  power  our  training 
should  proceed. 

Complex  conceptions  cannot  be  imposed  upon  a 
mind  incapable  of  receiving  them;  neither  can 
simple  truths.  Nothing  is  self-evident  save  to 
him  who  sees  it.  The  child  no  more  knows  that 
things  equal  to  the  same  thing  are  equal  to  each 
other,  until  he  sees  it  to  be  so,  than  he  knows 
that  yellow  and  blue  make  green.  He  sees  only 
that  which  he  has  the  power  to  see. 


8  THEORY    OF    ARITHMETIC. 

The  change  from  the  helplessness  of  the  babe 
to  the  power  of  the  child  of  six  is  a  constant 
miracle;  but  its  powers  are  still  relatively  feeble. 

Between  the  capacity  for  vague  perceptions 
and  for  framing  definite  mathematical  ideas  there 
are  many  intervening  stages.1  The  natural  ap- 
proach to  each  higher  thought-product  is  through 
the  lower  one,  which  is  its  necessary  antecedent. 

The  perception  of  equality  is  the  basis  of 
mathematical  reasoning,  —  a  condition  of  definite 
thinking.  But  a  child  sees  things  as  longer  or 
shorter,  larger  or  smaller,  before  he  is  able  to  see 
their  perfect  equality  or  exact  degree  of  inequality.2 
Until,  without  effort,  he  makes  such  discrimina- 
tions as  are  expressed  by  the  terms  long,  short, 
large,  small,  etc.,  he  is  not  ready  to  make  the 
discriminations  expressed  by  twice,  three  times, 
i  or  i. 

Analysis  dependent  upon  representative  power. — 

Exact   quantitative    relations    cannot    be    estab- 
lished without  analyzing.     Analysis  fixes  the  at- 

1 "  In  early  life  the  cerebral  organization  is  incomplete.  The 
period  necessary  for  completion  varies  with  the  race  and  with  the 
individual." — Prof.  Tyndall. 

2  "  The  conception  of  exact  likeness,"  remarks  Mr.  John  Fiske, 
"  is  a  highly  abstract  conception,  which  can  only  be  framed  after 
the  comparison  of  numerous  represented  cases  in  which  degree  of 
likeness  is  the  common  trait  that  is  thought  about." — Cosmic 
Philosophy,  vol.  ii.  p.  316. 


MATHEMATICS,  —  DEFI 

tention  in  turn  upon  each  part  rather  than  upon  the 
relation  of  the  compared  wholes.  When  the  pupil 
enters  upon  the  process  of  exact  comparison  he 
should  be  able  to  hold  each  term  of  the  comparison 
so  firmly  that  the  necessary  intrusion  of  a  common 
measure  will  not  efface  either  of  them.  Other- 
wise, the  operations  intended  to  throw  into  relief 
the  precise  relation  of  the  magnitudes  interpose 
as  a  cloud  to  render  the  relation  invisible. 

Place  a  measure  in  the  hands  of  a  pupil  and  set 
him  to  marking  off  spaces  on  this  and  that  and 
counting  them  before  he  is  ready  for  such  work, 
before  anything  has  been  done  to  induce  the  habit 
of  looking  from  one  magnitude  to  another,  and 
you  absorb  him  in  a  mechanical  process  which 
turns  the  thought  from  the  relational  element 
with  which  mathematics  deals.  He  may  write, 
"  The  door  is  8  feet  high,"  when  he  has  simply 
counted  8  spaces.  But  he  has  made  no  mathe- 
matical comparison,  observed  no  relation,  done 
little  which  tends  to  develop  power  to  think. 
If  we  ask  him  to  find  exact  relations  before  he 
has  sufficient  representative  power  to  bring  each 
term  of  the  comparison  into  consciousness  and 
approximate  its  relations  unaided,  the  probability 
is  that  the  relation  of  the  magnitudes  as  wholes 
will  not  be  seen  at  all. 

Premature  attempts  to  initiate  the  pupil  into 
the  ideas  of  mathematics  will  bewilder  him  with 


10  THEORY    OF    ARITHMETIC. 

the  mechanism  of  the  subject  and  create  a  condi- 
tion unfavorable  to  the  perception  of  mathemati- 
cal or  any  other  truth. 

"Not  only  is  it  true/'  says  Herbert  Spencer, 
"that  in  the  course  of  civilization  qualitative 
reasoning  precedes  quantitative  reasoning;  not 
only  is  it  true  that  in  the  growth  of  the  individual 
mind  the  progress  must  be  through  the  qualitative 
to  the  quantitative,  but  it  is  also  true  that  every 
act  of  quantitative  reasoning  is  qualitative  in  its 
initial  stages" 

Unity  of  subject.  —  The  teacher  must  be  clear 
as  to  what  characterizes  a  science.  Otherwise  the 
essential  may  be  lost  sight  of  in  the  subordinate, 
and  the  energy  of  the  pupil  wasted  in  the  effort  to 
unite  what  should  never  have  been  separated. 

A  living  apprehension  of  the  fact. that  mathe- 
matics deals  with  definite  relations  of  magnitude 
suggests  the  mode  of  beginning  the  study.  It 
suggests  the  need  of  creating  definite  ideas  ;  it 
forbids  presenting  things  as  isolated,  independent, 


absolute  in  themselves.  It  does  away  with  arti- 
ficial distinctions  between  a  fraction  and  an  in- 
teger, by  presenting  each  as  a  relation.  Thus  2  is 


MATHEMATICS,  —  DEFINITE   RELATIONS.  11 

the  relation  of  a  unit  to  another  half  as  large  ; 
and  one  half  is  the  relation  of  a  unit  to  another 
twice  as  large. 

A  relation  the  result  of  a  comparison.  —  To   be 

conscious  of  a  relation  means  more  than  to  be7 
conscious  of  the  terms  between  which  it  exists. 
We  may  think  of  the  taste  of  an  orange  or  of  a 
pear  without  connecting  them  in  any  way,  but  if 
we  are  considering  their  relative  sweetness  we 
must  bring  together  in  thought  the  taste  of  each ; 
a  comparison  must  take  place  before  we  can  assert 
that  one  is  sweeter  than  the  other.  So  we  may 
think  of  a  certain  line  as  one  yard,  of  another  as 
six  inches,  without  ability  to  assert  their  relative 
magnitude.  We  may  go  further  and  note  the  fact 
that  in  one  yard  there  are  six  six -inches,  and  still 
remain  without  any  appreciation  of  their  relative 
magnitude.  Before  we  can  assert  this,  the  intel- 
lectual act  which  brings  the  shorter  line  before 
the  mind  as  equal  to  one  sixth  of  the  longer  must 
take  place. 

We  cannot  meet  the  demands  of  mathematics 
by  observing  things  simply  as  distinct  and  sepa- 
rate. If  relations  are  to  come  into  consciousness, 
the  comparing  which  brings  them  there  must  take 
place.  An  example  may  make  this  more  clear. 
Suppose  the  magnitudes  a,  &,  c,  and  rf,  to  be  before 
the  child.  He  notes  likenesses  and  differences  in 


12  THEORY   OF   ARITHMETIC. 

them  just  as  he  does  in  colors,  leaves,  fruit,  or 
anything  to  which  he  attends.  Noting  d  and  a  he 
sees  that  d  is  greater  than  a,  that  a  is  less  than  d. 


He  has  made  comparisons  and  established  rela- 
tions, but  not  exact  relations.  These  relations  he 
expresses  by  the  indefinite  words  greater  and  less. 
If,  by  measuring,  he  effects  an  exact  comparison 
of  d  and  a,  he  needs  language  for  stating  that  the 
relation  of  d  to  a  is  3  ;  the  relation  of  a  to  d  is  J. 
He  may  call  a  |  and  d  1,  or  d  3  and  a  1 ;  or  he 
may  call  d  12  and  a  4,  but  their  relation  remains 
unchanged. 

The  thing  is  its  relations.  —  Comparing  c  with 
a  considered  as  1,  we  call  c  2.  Comparing  c  with 
d,  c  becomes  f,  yet  the  magnitude  c  has  not 
changed.  The  a  which  we  dealt  with  as  ^  when 
thought  of  in  relation  to  d,  as  ^  in  relation  to  c, 
we  call  1  when  compared  with  &,  or  with  any 
other  equal  magnitude. 


MATHEMATICS,  —  DEFINITE    RELATIONS.  13 

Just  as  the  child  learned  to  know  a  line  as  long 
in  comparison  with  another,  short  in  comparison 
with  a  third ;  to  call  a  day  warm  or  cold  accord- 
ing to  that  with  which  it  is  compared,  so  he  should 
learn  to  know  a  magnitude  as  2  when  seen  in  rela- 
tion to  a  magnitude  equal  to  its  ^ ;  to  see  the  same 
magnitude  as  3  or  5  or  ^  when  compared  with 
other  magnitudes. 

Means  of  comparing.  —  Effecting  an  exact  com- 
parison requires  analysis  and  synthesis,'  just  as 
every  act  does  which  results  in  a  judgment. 


In  order  to  discover  the  relation  of  4  to  6,  we 
may  separate  the  4  and  the  6  each  into  2's.  By 
the  analysis  (subtraction  or  division)  we  find  3 
2's  in  the  6  and  2- 2's  in  the  4.  Since  2  is  ^  of 
3- 2's,  we  infer  that  2  is  £  of  6.  (Why?)  In 
order  to  make  such  an  inference  we  must  see  that 
3*  2's  equal  6,  —  synthesis  (addition  or  multiplica- 
tion). 

From  successive  relations  of  equality  we  pass  to 
the  final  act  of  relating,  which  brings  2- 2's,  or  4, 
before  the  mind  as  equal  to  f  of  6.  The  final 
thought  is  not  of  the  4  nor  of  the  6,  nor  of  the 
relation  of  the  measuring  unit  to  either  ;  but  of 
the  relation  of  the  4  to  the  6.  In  no  case  have 
we  established  the  relation  sought  until  the  com- 


14 


THEORY    OF    ARITHMETIC. 


pared  wholes   are   brought  into  consciousness  in 
that  relation. 

Again,  suppose  we  wish  a  child  to  discover  the 
relation  which  exists  between  7  and  the  sum  of 
3  and  4. 


Mental  acts  must  take  place  showing  him  in 
the  7  two  magnitudes,  —  one  equal  to  4,  the  other 
equal  to  3,  —  and  bringing  the  7  again  before  the 
mind  as  one  quantity.  But  we  must  pass  be- 
yond these  steps  and  bring  the  seven  and  the 
sum  of  3  and  4  before  the  mind  in  the  relation 
of  equality.  A  judgment  of  relative  magnitude 
must  be  formed  which  unites  the  compared 
terms. 

Each  of  these  judgments,  like  every  other  judg- 
ment, is  the  product  of  analysis  and  synthesis — 
of  separating  and  uniting  ;  of  subtracting  and 
adding  ;  of  dividing  and  multiplying.  There  is 
no  real  synthesis  without  analysis,  no  addition 


MATHEMATICS,  —  DEFINITE    RELATIONS. 


15 


without  subtraction,  no  multiplication  without 
division. 

Conditions  of  comparing.  — 
In  comparing  there  must  be 
ideas  to  compare.  In  present- 
ing the  magnitude  7,  as  well  as 
3  and  4,  we  are  merely  meet- 
ing that  condition  of  thinking 
"  which  requires  that,  in  estab- 
lishing a  relation,  each  of  the 
compared  terms  must  be  pre- 
sent in  consciousness.  Through 
this  comparison  the  pupil  learns  to  know  7  in  one 
relation  ;  through  other  comparisons  he  will  enter 
into  fuller  knowledge  of  it. 

Meaning  of  a  word  depends  upon  experience.  — 

When  the  need  of  a  name  arises,  give  it.  The 
principle  is  the  same  whether  dealing  with  the 
qualitative  or  the  quantitative.  We  do  not  leave 
the  child  without  the  name  water  because  he  does 
not  know  the  elements  of  water.  We  tell  him  a 
certain  object  is  a  chair  long  before  he  has  a  com- 
plete idea  of  it.  As  his  surroundings  produce 
activity,  he  gradually  comes  to  know  special 
features  of  the  chair,  and  to  distinguish  arm- 
chairs, rocking-chairs,  etc. 

The  name  alone  can  avail  nothing ; l  but  when 

1  Language  attains  definiteness  for  the  individual  only  as  it  is 
associated  with  definite  ideas.     The  square  is  a  definite  figure  ; 


16  THEOBY    OF    AEITHMETIC. 

it  will  be  serviceable  in  focusing  the  attention,  in 
aiding  the  child  to  retain  his  grasp  of  a  thing,  and 
thus  in  facilitating  his  investigations,  it  should  be 
freely  given. 

Thought  and  expression  are  inseparable.  Words 
without  ideas  are  dead  ;  images  without  words 
are  elusive.  The  most  effective  method  of  mas- 
tering the  means  of  expression  in  mathematics, 
or  in  any  other  subject,  is  the  exercise  of  the  ' 
mind  upon  realities,  —  in  mathematics,  the  real- 
ities are  the  relations  of  magnitude. 

Using  divided  magnitudes  obscures  wholes,  weak- 
ens sense  of  coexistence.  —  By  presenting  divided 
magnitudes  (see  n  below),  we  destroy  the  wholes 
we  wish  compared,  and  call  upon  the  child  for 
a  synthesis  for  which  he  is  not  prepared.  The 
problem  does  not  require  him  to  make  a  com- 
parison of  the  magnitudes,  but  merely  to  count  the 
how  many.  We  force  upon  the  attention  isolated 
units  and  operations  for  which  the  mind  has  no 
need,  and  which,  by  being  thus  pushed  into  the 

but  the  child  may  handle  many  squares  and  repeat  the  definition 
of  a  square  many  times  without  any  feeling  of  its  definiteness.  If 
we  taught  the  child  to  say  that  the  sum  of  3  and  4  =  7,  without 
his  mentally  seeing  it  to  be  so,  we  should  be  presenting  symbols 
without  significance.  To  refuse  to  give  the  name  7  to  the  magni- 
tude in  this  particular  relation,  because  the  learner  is  not  fully 
conscious  of  the  meaning  of  the  term,  is  as  if  we  refused  to  allow 
a  child  to  talk  of  a  star  because  his  idea  of  it  is  not  that  of  the 
astronomer. 


MATHEMATICS,  —  DEFINITE    RELATIONS.  17 

foreground,  tend  only  to ,  intellectual  chaos.  A 
synthesis  not  accompanied  by  analysis  must  be 
artificial.  There  can  be  no  real  synthesis  without 
analysis. 


In  observing  n  (the  divided  magnitude)  does 
the  child  consider  the  relative  magnitude  of  the 
units  or  the  how  many?  In  comparing  c  (the 
undivided  magnitude)  with  d,  what  receives  the 
primary  attention,  the  how  many  or  the  relative 
size  ? 

The  child  grasps  a  dollar  or  a  dozen  as  a  unit, 
untroubled  by  its  composition.  So  it  should  grasp 
a  12,  a  17,  a  100,  a  J,  or  a  7.  So  it  will  if  you 
bring  them  into  consciousness  as  wholes.1  If  you 
wish  a  pupil  to  note  the  relation  between  the 
length  and  the  width  of  a  desk ;  or  between  a  12 
and  a  3 ;  a  1,200  and  a  300  ;  a  68  and  a  17;  or 

1 "  Now,  the  fact  is,  that  all  objects  of  apprehension,  including 
all  data  of  sense,  are  in  themselves,  i.e.  within  the  act  of  apprehen- 
sion, essentially  continuous.  They  become  discrete  only  by  being 
subjected,  arbitrarily  or  necessarily,  to  several  acts  of  apprehension, 
and  by  thus  being  severed  into  parts,  or  coordinated  with  other 
objects  similarly  apprehended  into  wholes." — J.  B.  Stallo, 


18  THEORY    OF    ARITHMETIC. 

a  |-  and  a  |,  what  are,  in  each  case,  the  wholes  to 
which  you  wish  him  to  attend  ? 

If,  instead  of  bringing  the  terms  of  the  compari- 
son before  the  mind  as  related  wholes,  we  require 
the  learner  to  begin  by  constructing  them  from 
the  parts,1  we  destroy  for  him  the  continuity  of 
the  magnitudes.  Consciousness  is  occupied  with 
a  succession  of  separate  units,  and  but  a  vague 
sense  of  the  relations  of  the  given  magnitudes  is 
awakened. 

Undivided  magnitudes;  use  induces  analysis  and 
synthesis.  —  It  must  not  be  supposed  that  the 
mere  use  of  undivided  magnitudes  will  insure 
the  perception  of  mathematical  relations;2  but  it 
fosters  such  perception.  It  is  a  condition  of  pres- 
entation in  accord  with  the  familiar  fact  that  the 

1 "  Where  the  parts  of  an  object  have  already  been  discerned, 
and  each  made  the  subject  of  a  special  discriminative  act,  we  can 
with  difficulty  feel  the  object  again  in  its  pristine  unity  ;  and  so 
prominent  may  our  consciousness  of  its  composition  be,  that  we 
may  hardly  believe  that  it  ever  could  have  appeared  undivided. 
But  this  is  an  erroneous  view,  the  undeniable  fact  being  that  any 
number  of  impressions,  from  any  number  of  sensory  sources,  falling 
simultaneously  on  a  mind  WHICH  HAS  NOT  YET  EXPERIENCED  THEM 
SEPARATELY,  will  fuse  into  a  single  undivided  object  for  that  mind." 
—  Wm.  James. 

2  The  material  provided  for  mental  nutrition  is  most  important. 
But  there  is  danger  of  relying  too  exclusively  upon  special  methods 
and  intrinsic  values.  Undue  reliance  upon  any  means  or  subject 
may  blind  us  to  the  fact  that  the  educational  process  is  not  going 
on  at  all. 


MATHEMATICS, DEFINITE    RELATIONS.  19 

mind  moves  from  the  whole  to  the  part  and  back 
again  to  the  whole;  that  it  analyzes  through  a 
desire  for  more  intimate  knowledge,  in  order  that 
it  may  reach  a  better  synthesis.  We  should  present 
as  wholes  the  magnitudes  whose  relations  we  wish 
established,  and  leave  the  way  open  for  those  suc- 
cessive acts  of  analysis  and  synthesis  by  which 
such  relations  are  established. 

Freeing  the  mind  from  the  concrete.  —  Noting  the 
same  relation  between  many  different  magnitudes 
tends  to  free  the  mind  from  the  concrete  and  the 
particular,  and  to  make  the  relations  the  objects 
of  thought. 

Thus  the  pupil  sees  magnitudes  differing  greatly 


in  size,  but  discovers  that  2  is  the  relation  not 
only  of  c  to  rf,  but  of  x  to  y,  of  a  to  &,  of  o  to 
m,  and  of  e  to  n\  he  notes  the  unlikeness  of  the 
separate  pairs,  the  likeness  of  their  relations;  he 


20  THEORY    OF    ARITHMETIC. 

is  asked  for  inference  after  inference  which  turns 
attention  to  the  ratio  of  the  units.1  Gradually 
he  learns  to  know  magnitudes  in  the  only 
way  that  they  can  be  known,  —  in  relation. 
The  simple  ratios  of  mathematics  become  real  to 
him. 

Giving  varying  names  to  the  units,  as  a  12  and 
a  6?  a  |  and  a  J,  a  100  and  a  50,  aids  in  separat- 
ing accidental  from  essential  relations,  and  in 
preventing  the  error  of  mistaking  the  relative  for 
the  absolute. 

Through  many,  very  many  experiences,  fitted 
for  developing  the  power,  he  becomes  able  to  dis- 
sociate the  relation  from  the  thing,  and  to  deal 
with  the  2,  the  3,  the  £,  the  |,  etc.,  as  uniform 
relations  upon  which  far-reaching  inferences  may 
be  based.2 

1 "  The  higher  processes  of  mind  in  mathematics  lie  at  the  very 
foundation  of  the  subject."  —  Sylvester. 

2  "  The  peculiarity  of  abstract  conceptions  is  that  the  matter  of 
thought  is  no  longer  any  one  object,  or  any  one  action,  but  a  trait 
common  to  many  ;  and  it  is,  therefore,  only  when  a  number  of 
distinct  objects  or  relations  possessing  some  common  trait  can  be 
represented  in  consciousness,  that  there  becomes  possible  that 
comparison  which  results  in  the  abstraction  of  the  common  trait 
as  the  object  of  thought."  — John  Fiske. 

"  The  development  of  ideas  is  the  slow,  gradual  result  of  contin- 
uous judgment."  —  Francis  Bowen. 

"  What  is  associated  now  with  one  thing  and  now  with  another 
tends  to  become  dissociated  from  either,  and  to  grow  into  an 
object  of  abstract  contemplation  by  the  mind."  —  Wm.  James. 


MATHEMATICS,  —  DEFINITE    RELATIONS.  21 

Inference  must  succeed  perception. — The  import- 
ance of  bringing  simple  basic  ratios  definitely  into 
consciousness  is  better  understood  when  we  look 
beyond  them.1 

The  development  of  mathematics  within  the 
mind,  and  the  development  of  the  mind  by  means 
of  mathematics,  are  alike  impossible  without  that 
thinking,  relating,  reasoning,  by  which  the  mind 
"produces  from  what  it  receives."  From  the 
beginning  we  must  address  the  mind  and  not  one 
function ;  give  opportunity  for  inference  to  suc- 
ceed perception.  By  unduly  crowding  the  sensing 
and  recording  of  ratios  we  may  so  handicap  the 
mind  that  it  cannot  move.  As  law  or  principle 
serves  the  man  of  science,  so  each  simple  truth 
should  serve  the  child  in  lighting  the  way  to 
other  truths. 

By  means  of  perceived  relations  we  must  pass 
to  the  inferring  of  relations.  For  example,  it  is 
not  enough  for  the  child  to  see  the  relation  of  d  to 
a  and  of  a  to  d.  From  these  perceptions  he  must 
infer  other  relations.  Rightly  taught,  such  infer- 
ences as  that  the  weight  of  d  equals  3  times  the 

1 "  And  if  we  neglect  to  educe  the  fundamental  conception 
on  which  all  his  ulterior  knowledge  must  depend  we  not  only 
sow  the  seed  of  endless  obscurity  and  perplexity  during  all  his 
future  advance  in  this  science,  but  we  also  weaken  his  reasoning 
habits  .  .  .  and  thus  make  our  mathematical  discipline  produce, 
not  a  wholesome  and  invigorating  but  a  deleterious  and  pervert- 
ing effect  upon  the  mind." — Whewell. 


22  THEORY    OF    ARITHMETIC. 

weight  of  a ;  that  3  times  as  many  inch  cubes  can 
be  cut  from  d  as  from  a ;  that  d  will  yield  3  times 
as  much  ashes  as  a;  that  the  cost  of  d  equals  3 
times  the  cost  of  a ;  that  the  weight  of  a  equals  ^ 
the  weight  of  d\  that  a  will  yield  \.  the  amount  of 


ashes  that  d  will  yield ;  that  the  cost  of  a  equals  \ 
the  cost  of  d,  will  follow  naturally  and  readily  upon 
the  perception  of  the  ratio  of  d  to  a  and  of  a  to 
d.  They  will  never  follow  without  the  generating 
conditions,  and  the  generating  conditions  are  per- 
ceptions of  exact  relations.  Upon  these  equations, 
made  known  by  the  activity  of  the  mind  upon  the 
magnitudes  themselves,  all  mathematical  deduc- 
tion depends. 

We  frequently  hear  it  said,  "  Is  it  not  a  proof 
that  the  child  sees  the  conditions  when  he  says  that 
d  will  cost  3  x  if  a  cost  x  ? "  Or,  if  this  is  not 
enough,  he  can  tell  you  that,  "  Because  d  is  3 
times  a,  etc."  Experience  shows,  however,  that 
many  pupils  who  can  do  all  this  will  tell  you 
a  little  later  that  it  will  take  3  boys  3  times  as 
long  to  do  a  piece  of  work  as  it  will  one  boy  ; 


MATHEMATICS,  —  DEFINITE    RELATIONS.  23 

that  the  weight  of  a  2 -inch-  cube  is  twice  that  of 
an  inch  cube.  As  they  advance,  their  seeming 
inaptitude  for  mathematics  becomes  more  marked. 
Why  is  this  ?  Because  a  wrong  direction  was 
given  to  the  mind  in  the  beginning;  because 
mechanical  processes  for  securing  results  were 
substituted  for  those  experiences  which  create 
ideas  of  equality  and  of  exact  ratio ;  because 
using  objects  merely  to  teach  children  to  count 
and  to  manipulate  numbers,1  instead  of  presenting 
them  in  such  a  way  as  to  attract  attention  to 
their  relative  magnitude,  leaves  the  mind  without 
any  basis  for  the  deductions  which  are  demanded. 
Out  of  number  nothing  comes  save  number.  If 
we  ask  for  conclusions  concerning  quantity  we 
must  see  that  the  mind  possesses  a  basis2  for  those 
conclusions.  The  indispensable  groundwork  of 
reasoning  is  the  definite  mental  representation  of 
the  relation  upon  which  an  inference  rests;  and 
mathematical  inferences  rest  upon  ratios. 

Clear  imaging ;  clear  thinking ;  correct  conclusion. 

—  The  material   upon  which  the  mind   can   act 
from   time   to   time    depends   upon   its    growing 

1 "  It  would  indicate  a  radically  false  idea  of  number  to  wish 
to  employ  it  in  establishing  the  elementary  foundations  of  any 
science  whatever  ;  for  on  what  would  the  reasoning  in  such  an 
operation  repose  ?  "  —  Comte. 

2  "  The  attempt  to  found  the  science  of  quantity  upon  the 
science  of  number  I  believe  to  be  radically  wrong  and  educa- 
tionally mischievous." — Win.  K.  Clifford. 


24  THEORY    OF    ARITHMETIC. 

power  to  represent  in-  thought  the  conditions  upon 
which  conclusions  follow. 

Pupils  accept  the  statement  that  it  will  take 
twice  as  long  to  paint  a  2-inch  square  as  a  1-inch 
square  because  they  do  not  represent  the  squares 
mentally. 

If  the  pupil  has  been  trained  so  that  it  is  his 
habit  to  make  the  necessary  mental  representa- 
tions he  will  see  for  himself  that  if  x  is  the  num- 
ber of  yards  of  carpet  1J  yards  wide  required  for 
a  floor,  2  x  yards  f  of  a  yard  wide  will  be  needed. 
No  wordy  explanation  will  be  required.  Yet  pupils 
fail  constantly  in  such  simple  exercises.  They 
cannot  make  comparisons,  because  they  have  in 
their  minds  no  images  of  the  things  they  are  to 
compare.  They  cannot  deduce  from  symbols  the 
relations  of  reals.1  Asking  pupils  to  reason  about 
things  which  they  do  not  see  mentally,  is  asking 
the  impossible  and  can  only  lead  to  confusion  and 
discouragement.  The  power  of  representative 
thought,  of  imaging,  underlies  all  intellectual 
progress,  and  we  cannot  prepare  the  mind  for 
abstract  thought  without  developing  this  power. 

Mathematics  deals  with  realities.  —  However 
divergent  may  be  the  lines  of  mathematical 

1  "  How  accurate  soever  the  logical  process  may  be,  if  our  first 
principles  be  rashly  assumed,  or  if  our  terms  be  indefinite  or 
ambiguous,  there  is  no  absurdity  so  great  that  we  may  not  be 
brought  to  adopt  it."  —  Dugald  Stewart. 


MATHEMATICS,  —  DEFINITE    KELATIONS.  25 

thought,  their  beginnings  are  sensible  intuitions,  — 
that  is,  the  ideas  of  magnitude  must  be  based 
on  perceptions;  and  however  long  the  line,  its 
extension  is  in  all  cases  by  means  of  successive 
acts  of  comparing  and  inferring. 

Sylvester  finds  that  "  The  study  of  mathematics 
is  unceasingly  calling  forth  the  faculties  of  obser- 
vation and  comparison ;  that  it  has  frequent 
recourse  to  experimental  trial  and  verification; 
and  that  it  affords  a  boundless  scope  for  the 
highest  efforts  of  imagination.  ...  I  might  go 
on,"  he  says,  "  piling  instance  upon  instance  to 
show  the  paramount  importance  of  the  faculty  of 
observation  to  the  process  of  mathematical  dis- 
covery." 

"  Mathematics,"  says  Mr.  Lewes,  "  is  a  science 
of  observation,  dealing  with  reals,  precisely  as  all 
other  sciences  deal  with  reals.  It  would  be  easy 
to  show  that  its  method  is  the  same."  The  reals 
are  the  relations  of  magnitude. 

The  order  of  truth  changes ;  the  mental  action 
which  embraces  it  remains  the  same.  We  note 
the  likenesses  of  two  leaves  or  the  exact  likenesses 
of  two  magnitudes  ;  in  each  case  we  have  a  basis 
for  inference  obtained  by  comparing.  When  we 
turn  to  exact  likenesses,  we  enter  the  domain  of 
mathematics. 

Objects  unfitted  to  awaken  mathematical  ideas.  — 
Were  we  concerned  simply  with  the  number  of 


26  THEORY   OF   ARITHMETIC. 

things,  beans,  shoe-pegs,  shells,  leaves,  pebbles, 
chairs,  or  the  legs  of  frogs  might  serve  as  well 
as  anything.  But  mere  numerical  equality  will 
not  serve  as  a  basis  for  mathematical  reasoning ; 
-exact  results  cannot  be  founded  upon  it. 

Dealing  with  units,  without  regard  to  their 
equality  or  inequality  ;  considering  them  only  as 
distinct  things;  and  reaching  results  true  only 
numerically,  has  been  called  the  indefinite  calculus; 
but  the  indefinite  calculus  furnishes  no  basis  for 
mathematical  reasoning.  If  arithmetic  is  made 
merely  a  means  of  teaching  number,  and  opera- 
tions with  number,  it  should  receive  but  brief 
time  in  the  common-school  course.  Very  little  of 
it  will  suffice  for  the  ordinary  vocations  of  life. 
The  cases  in  which  mere  numerical  relations  are 
considered  are  so  simple  as  scarcely  to  stir  the 
mind.1 

A  superficial  knowledge  of  mathematics  may 
lead  to  the  belief  that  this  subject  can  be  taught 
incidentally,  and  that  exercises  akin  to  counting 
the  petals  of  a  flower  or  the  legs  of  a  grasshopper 
are  mathematical.  Such  work  ignores  the  funda- 
mental idea  out  of  which  quantitative  reasoning 

1  In  regard  to  the  how  many,  to  work  which  does  not  deal 
with  definite  relations,  Comte  said,  "  This  will  never  be  more 
than  a  point,  so  to  speak,  in  comparison  with  the  establishment 
of  relations  of  magnitude  of  which  mathematical  science  essen- 
tially consists.  ...  In  this  point  of  view,  arithmetic  would 
disappear  as  a  distinct  section  in  the  whole  body  of  mathematics." 


MATHEMATICS,  —  DEFINITE    RELATIONS.  27 

grows — the  equality  of  magnitudes.1  It  leaves 
the  pupil  unaware  of  that  relativity  which  is  the 
essence  of  mathematical  science.  Numerical  state- 
ments are  frequently  required  in  the  study  of 
natural  history,  but  to  repeat  these  as  a  drill  upon 
numbers  will  scarcely  lend  charm  to  these  studies, 
and  certainly  will  not  result  in  mathematical 
knowledge. 

Vague  ideas  of  the  unlikeness  of  a  rhomboid, 
a  square,  and  a  trapezium  may  be  gained  by  count- 
ing them,  and  so  may  vague  ideas  of  the  relations 
of  magnitude.  If  definite  ideas  of  color,  form,  or 
weight  come  from  counting  and  learning  tables, 
then  definite  ideas  of  quantitative  relations  may 
come  in  the  same  way. 

Turning  from  the  numbering  of  things  to  their 
mathematical  comparison,  we  see  at  once  why 
plants  and  animals  are  not  well  adapted  for  our 
purpose.  In  them,  that  which  is  material  is 
obscured  by  that  which  is  irrelevant.2  It  is  diffi- 
cult for  the  undeveloped  mind  to  view  these  objects 

1  "  Equations  constitute  the  true  starting  point  of  arithmetic." 
—  Comte. 

"  The  fundamental  ideas  underlying  all  mathematics  is  that  of 
equality."  —  Herbert  Spencer. 

2  "  The   visible   figures   by   which    principles    are    illustrated 
should,  so  far  as  possible,  have  no  accessories.     They  should  be 
magnitudes  pure  and  simple,  so  that  the  thought  of  the  pupil  may 
not  be  distracted,  and  that  he  may  know  what  feature  of  the 
thing   represented   he  is  to   pay  attention  to."  —  Committee   of 
Ten." 


28  THEORY   OF   ARITHMETIC. 

in  their  mathematical  aspect.  Their  differences 
in  magnitude  are  not  easily  appreciated  by  the 
senses.  Their  exact  measurement  is  not  easy. 
They  lend  themselves  to  accurate  imaging  far  less 
readily  than  simple  magnitudes,  and  do  not  result 
in  those  mental  states  which  would  be  created 
were  mathematical  relations  brought  conspicu- 
ously and  impressively  into  the  pupils'  experiences, 
That  mathematics  enters  into  other  sciences  is 
understood.  The  fruitfulness  of  physics  for  the 
teacher  of  mathematics  is  apparent.  Advancing 
science  is  constantly  making  more  clear  the  inter- 
dependencies  of  the  various  sciences.  Each  aids 
in  the  development  of  the  others.1  But  it  does 

1  "  Although  each  science  throws  its  light  on  every  other, 
owing  to  the  interdependence  of  phenomena  and  the  community 
of  consciousness,  yet  .  .  .  phenomena  are  independent  not  less 
than  interdependent.  Mathematics  cannot  receive  laws  from 
chemistry,  nor  physics  from  biology  ;  the  phenomena  studied  in 
each  are  special."  —  Lewes. 

"  This  unification  of  all  the  modes  of  existence  by  no  means 
obliterates  the  distinction  of  modes,  nor  the  necessity  of  under- 
standing the  special  characters  of  each.  ...  If  we  recognize  the 
one  in  the  many,  we  do  not  thereby  refuse  to  admit  the  many  in 
the  one"  —  Lewes. 

"  Sciences  are  the  result  of  mental  abstraction,  being  the 
logical  record  of  this  or  that  aspect  of  the  whole  subject-matter  of 
knowledge.  As  they  all  belong  to  one  and  the  same  circle  of 
objects,  they  are  one  and  all  connected  together  ;  as  they  are  but 
aspects  of  things,  they  are  severally  incomplete  in  their  relation 
to  the  things  themselves,  though  complete  in  their  own  idea  and 
for  their  own  respective  purposes  ;  on  both  accounts  they  at  once 
need  and  subserve  one  another."  —  Cardinal  Newman. 


MATHEMATICS,  —  DEFINITE    RELATIONS.  29 

not  follow  that  different  .classes  of  ideas  will  be 
equally  excited  by  the  same  objects. 

The  result  of  trying  to  call  forth  mathematical 
ideas  by  means  of  phenomena  whose  exact  meas- 
urement is  beyond  the  power  of  the  pupil,  is  very 
similar  to  the  result  when  no  pretence  is  made  of 
founding  deduction  upon  perception.  Why  should 
it  not  be  ?  In  neither  case  do  mathematical  rela- 
tions come  definitely  into  consciousness. 

What  objects  will  excite  definite  ideas  ?  —  Things 
whose  exact  relations  can  be  most  readily  seen ; 
things  which  can  be  most  accurately  imaged  and 
exactly  compared ;  things  which  tend  most  to 
excite  definite  intuitions  and  to  result  in  definite- 
ness  of  mind,,  should  be  given  precedence  in 
elementary  instruction  in  mathematics. 

Comte  observes,  "  The  only  comparisons  capable 
of  being  made  directly,  and  which  could  not  be 
reduced  to  any  others  more  easy  to  effect,  are 
the  simple  comparisons  of  right  lines."  This  is 
apparent  to  whoever  gives  thought  to  the 
matter.1 

1 "  On  tracing  them  back  to  their  origins,  we  find  that  the  units 
of  time,  force,  value,  velocity,  etc.,  which  figures  may  indiscrimi- 
nately represent,  were  at  first  measured  by  equal  units  of  space. 
The  equality  of  time  becomes  known  either  by  means  of  the  equal 
spaces  traversed  by  an  index,  or  the  descent  of  equal  quantities 
*  (space-fulls)  of  sand  or  water.  Equal  units  of  weight  were 
obtained  through  the  aid  of  a  lever  having  equal  arms  (scales). 


30  THEORY   OF   ARITHMETIC. 

Since  the  measurement  of  all  magnitudes  is 
reducible  to  measurements  of  linear  extension,  and 
since  the  comparison  of  linear  units  alone  reveals 
that  perfect  equality  upon  which  the  science  of 
mathematics  is  built,  —  since  by  such  comparisons 
and  only  by  them  do  we  obtain  the  original 
materials  of  mathematical  thought,  since  these 
experiences  alone  give  rise  to  those  abstract  con- 
ceptions which  enable  us  to  use  numbers  intelli- 
gently,— it  follows  that  definite  magnitudes  should 
furnish  the  objective  stimulus  in  laying  a  basis  for 
mathematical  knowledge.  Out  of  ratios  estab- 
lished by  comparing  right  lines  the  ratios  of 
surfaces  and  solids  are  inferred,  and  also  the 
quantitative  relations  of  units  of  value,  force,  and, 
in  short,  of  all  other  magnitudes. 

The  problems  of  statics  and  dynamics  are  primarily  soluble,  only 
by  putting  lengths  of  lines  to  represent  amounts  of  forces.  Mer- 
cantile values  are  expressed  in  units  which  were  at  first,  and 
indeed  are  still,  definite  weights  of  metal ;  and  are,  therefore,  in 
common  with  units  of  weight,  referable  to  units  of  linear  exten- 
sion. Temperature  is  measured  by  the  equal  lengths  marked 
alongside  a  mercurial  column.  Thus,  abstract  as  they  have  now 
become,  the  units  of  calculation,  applied  to  whatever  species 
of  magnitude,  do  really  stand  for  equal  units  of  linear  extension, 
and  the  idea  of  coextension  underlies  every  process  of  mathemat- 
ical analysis.  Similarly  with  coexistence.  Numerical  symbols 
are  purely  representative  ;  and  hence  may  be  regarded  as  having 
nothing  but  a  fictitious  existence."  —  Spencer,  Principles  of  Psy- 
chology, vol.  ii.  p.  38. 

"  Whenever  I  went  far  enough  I  touched  a  geometrical 
bottom." — Prof.  Sylvester,  Address  British  Association,  1869. 


MATHEMATICS,  —  DEFINITE    RELATIONS.  31 

Means  of  passing  beyond  the  range  of  percep- 
tion.—  It  is  the  definite  relations  of  magnitudes 
established  by  means  of  solids,  surfaces,  and  lines, 
that  enable  us  to  conceive  or  interpret  the  rela- 
tions of  quantities  which  cannot  be  brought  within 
the  range  of  perception.  The  ratios  which  we 
actually  see  are  few,  but  out  of  these  grows  the 
science  of  mathematics. 

These  primary  relations,  then,  should  be  so 
repeatedly  felt,  so  ingrained,  that  they  will  become 
elements  in  the  mental  life.  This  is  possible  only 
by  confronting  the  pupil  again  and  again  with  the 
conditions  which  force  upon  him  the  methods  and 
ideas  of  mathematics.  He  should  become  so  iden- 
tified with  the  kind  of  relations  dealt  with,  that 
the  abstract  terms  in  which  he  afterwards  reasons 
will  be  truly  representative.  Otherwise,  he  will 
restrict  and  misapply  them.  It  is  the  certainty 
of  the  seen  that  makes  us  rationally  certain  of  the 
unseen. 

The  basis  of  drills  the  perception  of  relations.  — 
It  is  well  understood  that  the  use  of  language 
must  become  automatic  if  the  mind  is  to  move 
freely  in  the  discovery  of  laws  and  principles. 

How  is  this  needful  familiarity  with  the  means 
of  making  quantitative  comparisons  to  be  provided 
for?  Not,  certainly,  by  treating  the  means  as 
though  it  were  the  end  ;  not  by  forcing  premature 


32  THEORY    OF    ARITHMETIC. 

drill  upon  tables  and  routine  work  in  combining 
and  separating  symbols.  This  is  to  ignore  mathe- 
matics, to  ignore  natural  sequences,  both  within 
and  without  the  mind.  Its  tendency  is  to  prevent 
energy  from  rising  to  that  higher  kind  of  power  of 
which  an  intelligent  being  is  capable. 

The  drills  should  harmonize  with  the  dominant 
idea  of  the  subject  and  meet  the  conditions  which 
favor  retention  without  interfering  with  growth. 


In  his  observing  and  comparing,  the  pupil  has 
dealt  with  the  ratios  2,  3,  4,  etc.  He  has  seen 
that  the  ratio  of  4  to  2  equals  the  ratio  of  6  to  3, 
of  8  to  4,  of  10  to  5,  of  £  to  £,  etc.  We  bring 
these  equal  ratios  together  in  the  same  table  and 
associate  them  in  his  mind.  Making  the  common 
thing,  the  ratio,  prominent,  unifies  the  work  and 
relieves  the  memory.  Grouping  like  ratios  in  the 
drills  is  analogous  to  the  grouping  required  in 
solving  problems.  Thus,  the  pupil  sees  that  the 
relation  of  the  cost  of  6  acres  to  the  cost  of  2 
acres  is  equal  to  the  relation  of  their  areas.  From 
one  truth  he  passes  to  another,  and  brings  the 
differing  ideas  into  unity.  The  drills  should  em- 
phasize this  sense  of  likeness  in  the  midst  of 


MATHEMATICS, DEFINITE    RELATIONS.  33 

difference  without  interfering  with  the  flexibility 
of  the  mind. 

Drill  work  should  be  a  means  of  increasing  men- 
tal power  by  training  the  eye  to  quickness  and 
accuracy,  and  the  mind  to  attend  closely  and  image 
vividly. 

In  every  exercise  the  first  thing  to  secure  is  a 
clear  mental  picture.  When  the  pictures  are  dis- 
tinct, work  for  rapidity.  What  is  to  be  recog- 
nized at  sight  should  be  taken  in  through  the  eye.1 
The  visual  image  will  be  dimmed  and  blurred,  and 

1 "  A  common  error,  into  which  beginners  are  apt  to  fall,  is  to 
try  to  combine,  and  therefore  to  confuse,  the  two  methods  of 
remembering,  by  sight  and  by  sound."  —  Dr.  M.  Granville. 

"  When  a  child  first  sees  a  thing,  it  takes  it  in  by  the  eye  ; 
when  it  first  hears  a  thing,  it  takes  it  in  by  the  ear  ;  in  each  case 
the  whole  mind  is  concentrated  on  the  sensation,  which,  as  Dr. 
Carpenter  says,  *  is  the  natural  state  of  the  infant.'  But  as  soon 
as  education  begins,  all  this  is  changed,  and  the  mind,  instead  of 
being  concentrated  upon  one  thing,  is  distracted  by  several." — 
Kay. 

"  We  must  attend  to  the  formation  of  the  original  impression 
.  .  .  and  recall  it  in  its  entirety  afterwards."  —  Kay. 

"  Nothing  needs  more  to  be  insisted  on  than  that  vivid  and 
complete  impressions  are  all-essential."  —  Herbert  Spencer. 

"  There  can  be  no  doubt  as  to  the  utility  of  the  visualising 
faculty  when  it  is  duly  subordinated  to  the  higher  intellectual 
operation.  A  visual  image  is  the  most  perfect  form  of  mental 
representation  wherever  the  shape,  position,  and  relations  of 
objects  in  space  are  concerned." —  F.  Galton. 

"  The  more  completely  the  mental  energy  can  be  brought  into 
one  focus,  and  all  distracting  objects  excluded,  the  more  powerful 
will  be  the  volitional  effort."  —  Dr.  Carpenter. 


34  THEORY   OF    ARITHMETIC. 

hence  imperfectly  remembered,  if  we  attempt  to 
call  the  ear  into  action  at  the  same  time  that  we 
address  the  eye. 

The  way  not  to  succeed  in  memorizing  the  tables 
is  to  repeat  so  many  different  impressions  in  the 
same  exercise  that  none  of  them  are  distinct ;  to 
confuse  eye  and  ear  training ;  to  make  the  work 
so  difficult  that  it  cannot  be  done  easily  and 

"  It  is  a  matter  of  common  remark  that  the  permanence  of  the 
impression  which  anything  leaves  on  the  memory  is  proportioned 
to  the  degree  of  attention  which  was  originally  given  it."  —  D. 
Stewart. 

"  Most  persons  find  that  the  first  image  they  have  acquired  of 
any  scene  is  apt  to  hold  its  place  tenaciously. " —  F.  Galton. 

"  The  habit  of  hasty  and  inexact  observation  is  the  foundation 
of  the  habit  of  remembering  wrongly."  —  Dr.  Maudsley. 

"  No  ideas  can  long  be  retained  in  the  memory  which  are  not 
deeply  fixed  by  repetition."  —  Joseph  Payne. 

"  The  leading  principle  is  to  learn  very  little  at  a  time,  not  in 
a  loose,  careless  way,  but  perfectly."  —  P.  Prendergast. 

"A  few  such  items  must  be  memorized  and  reviewed  daily, 
adding  a  small  increment  to  the  list  as  soon  as  it  has  become 
perfectly  mastered."  —  W.  T.  Harris. 

"  We  usually  attempt  to  master  too  much  at  once,  and  hence 
the  impressions  formed  in  the  mind  lack  clearness  and  distinct- 
ness." —  Kay. 

"  All  improvement  in  the  art  of  teaching  depends  on  the  atten- 
tion that  we  give  to  the  various  circumstances  that  facilitate 
acquirement  or  lessen  the  number  of  repetitions  for  a  given 
effect."  —  Prof.  Bain. 

"  It  is  not  enough  that  impressions  be  received  ;  they  must  be 
fixed,  organically  registered,  conserved ;  they  must  produce  per- 
manent modifications  in  the  brain.  .  .  .  This  result  can  depend 
only  on  nutrition."  —  Th.  Ribot. 


MATHEMATICS,  —  DEFINITE   RELATIONS.  35 

quickly  ;  to  drill  once  or  twice  a  month  ;  and  to 
prolong  the  exercise  until  the  power  of  attention 
is  exhausted. 

The  way  to  succeed  is  to  develop  vivid  mental 
pictures,  and  to  fix  these  pictures  by  bringing 
them  again  and  again  before  the  mind. 

Briefly  summarized,  we  may  say  :  Reasoning 
in  arithmetic  establishes  equality  of  relations ; 
reasoning  in  any  subject,  equality  or  likeness  of 
relations. 

We  know  magnitudes  only  in  relation  ;  and  the 
purpose  of  mathematical  science  is  to  establish 
definite  relations  between  magnitudes.  The  funda- 
mental operation  is  comparison.  Out  of  the  rela- 
tions established  by  comparison  grow  inferences. 

Only  through  the  activity  of  the  mind  in  observ- 
ing and  comparing  can  those  equations  be  formed 
which  are  the  groundwork  of  reasoning,  the  basis 
of  advance  from  relations  seen  to  relations  which 
lie  beyond  the  range  of  perception.1 

That  quantity  is  a  ratio  between  terms  which 
are  themselves  relative  ;  that  mathematics  is  not 

luThe  domain  of  the  senses,  in  nature,  is  almost  infinitely 
small  in  comparison  with  the  vast  region  accessible  to  thought 
which  lies  beyond  them.  .  .  .  By  means  of  data  furnished  in  the 
narrow  world  of  the  senses,  we  make  ourselves  at  home  in  other 
and  wider  worlds,  which  are  traversed  by  the  intellect  alone.  .  .  . 
We  never  could  have  measured  the  waves  of  light,  nor  even 
imagined  them  to  exist,  had  we  not  previously  exercised  ourselves 
among  the  waves  of  sound."  —  Prof.  Tyndall. 


36  THEORY    OF    ARITHMETIC. 

concerned  with  things  as  separate  and  absolute  ; 
that  it  deals  only  with  relations,  are  truths  which 
have  often  been  pointed  out,  but  which  the  work 
of  the  school  shows  to  be  felt  by  few. 

In  the  light  of  these  ideas,  those  arbitrary 
divisions,  so  fatal  to  the  continuous  unfolding  of 
thought,  are  seen  to  belong  to  our  language  and 
our  schemes  of  study,  rather  than  to  the  subject. 

Make  definite  relations  the  basis,  and  the  integer 
and  the  fraction  are  each  seen  as  a  ratio  ;  geo- 
metry, arithmetic,  and  algebra  merge  insensibly 
into  one  another.  With  definite  relations  as  the 
center,  it  becomes  clear  that  if  we  would  teach 
mathematics,  and  not  the  mere  mechanism  of  the 
subject,  we  must  look  to  the  development  of  the 
representative  and  comparative  powers.  Only  thus 
can  we  lift  arithmetic  from  a  matter  of  memory, 
routine,  and  formula  to  its  rightful  place  as  a 
means  of  enlarging  the  mind. 


PEIMAET    ARITHMETIC. 


FIRST   STEPS.  — SENSE   TRAINING. 


Finding  solids.  —  Place  spheres,  cubes,  cylinders,  and 
other  forms  of  various  sizes  in  different  parts  of  the  room 
where  the  children  can  find  them. 

Show  a  sphere  to  the  pupils.     Ask  : 

1.  What  is  this? 

Find  other  balls  or  spheres. 
Find  a  larger  sphere  than  this.     Find  smaller 
ones. 

2.  Name  objects  like  a  sphere.     Example  ;  An 
orange  is  like  a  sphere. 


38  PRIMARY    ARITHMETIC. 

3.  What  is  the  largest  sphere  that  you  have, 
seen? 

What  is  one  of  the  smallest  spheres  that  you 
have  seen? 

4.  To-morrow  tell   me   the  names  of   spheres 
that  you  see  when  going  from  school  and  at  home. 

Ask,  to-morrow,  for  the  names  of  the  objects  and  where 
they  were  seen. 

5.  What    is    the    largest    sphere    you    found? 
What  is  the  smallest? 

Review  and  work  in  a  similar  way  with  other  solids. 

"He  should  at  first  gain  familiarity  through  the  senses 
with  simple  geometrical  figures  and  forms,  plane  and  solid  ; 
should  handle,  draw,  measure,  and  model  them  ;  and  should 
gradually  learn  some  of  their  simpler  properties  and  rela- 
tions." —  Committee  of  Ten. 

Children  recognize  objects  similar  in  form,  color,  etc., 
before  they  desire  or  have  the  ability  to  express  what  they 
see. 

Until  a  child  can  readily  select  a  form  he  is  not  ready 
to  make  a  statement  of  what  he  has  found.  Let  the 
approach  to  telling  be  through  doing  ;  through  the  activity 
of  the  pupil  in  discriminating  and  relating. 

The  teacher,  and  such  pupils  as  are  able,  should  use  the 
proper  terms,  so  that  pupils  who  have  not  heard  the  terms 
may  learn  to  apply  them.  Children  can  discover  like- 
nesses and  differences  —  relations  —  but  not  the  terms  in 
which  they  are  expressed.  They  should  learn  the  terms 
unconsciously  by  living  in  an  atmosphere  where  they  are 
used.  Since  we  think  most  easily  in  the  names  we  have 
first  and  most  familiarly  associated  with  a  thing,  the  right 


PRIMARY    ARITHMETIC.  39 

term  should  be  used  from,  the  beginning.  Providing  fitly  for 
expressing  is  an  important  means  of  arousing  self-activity. 
The  different  exercises  are  to  be  continued  from  day  to 
day,  as  the  growing  interest  and  powers  of  the  child 
suggest,  and  until  there  is  skill  in  performing  and  ease  in 
expressing.  The  teacher  should  know  the  condition  of  the 
pupil's  mind.  His  expression  is  the  index  to  his  mental 
state.  Avoid  anything  which  will  tend  to  substitute 
mechanical  expression  for  real  expression.  Any  form 
which  is  not  the  outgrowth  of  what  is  within,  which  is  not 
the  genuine  product  of  free  activity,  will  mislead  the 
teacher  and  weaken  the  child. 

"Forms  which  grow  round  a  substance  will  be  true, 
good  ;  forms  which  are  consciously  put  round  a  substance, 
bad.  I  invite  you  to  reflect  on  this."  —  Carlyle. 

Finding  colors.  —  Tests  in  color  should  be  given  before 
the  more  formal  work  suggested  below.  For  example  : 
Group  cards  of  the  same  color  and  threads  of  worsted.1 

Provide  ribbons,  worsted,  cards,  etc.,  of  different  colors, 
to  be  found  by  pupils  when  looking  for  a  particular  color. 

Pin  or  paste  squares  of  standard  red  and  orange  where 
they  can  be  seen.  Pin  the  red  above  the  orange. 

1.  Find  things  in  the  room  of  the  same  color 
as  the  red  square.  What  things  can  you  recall 
that  are  red  ? 

1  These  exercises  are  not  to  teach  color,  but  are  to  train  pupils 
to  visualize,  to  attend,  to  compare,  and  to  secure  greater  freedom 
in  expressing  through  noting  different  relations.  All  pupils  need 
such  work  before  beginning  the  usual  studies  of  the  primary 
school.  They  lack  needful  elementary  ideas,  which  must  be 
obtained  through  the  senses.  The  range  of  the  perceptions  needs 
to  be  widened. 


40  PRIMARY   ARITHMETIC. 

2.  Look  at  the  orange  square.     Find  the  same 
color  elsewhere  in  the  room.     Recall  objects  that 
have  this  color. 

3.  Close  the  eyes,  and   picture  or  image   the 
red  square.     Now  the  orange  square. 

4.  Which   square   is   above?      Which    below? 
Name  the  two  colors. 

5.  To-morrow  bring  something  that  is  red  and 
something  that  is  orange.     Also  tell  the  names  of 
orange  or  red  objects  that  you  see  in  going  to  and 
from  school. 

Pin  or  paste  a  square  of  yellow  below  the  orange. 

1.  Look  at  the  yellow.     Find  the  same  color  in 
the  room.     Recall  objects  having  this  color. 

2.  Look  at  the  red,  then  the  orange,  then  the 
yellow.     Close  the  eyes  and   picture   the   colors 
one  after  another  in  the  same  order. 

Cover  the  squares. 

3.  Which  color  is  at  the  top ?     At  the  bottom? 
In  the  middle  ? 

4.  Name  the  three,  beginning  at  the  top.    Name 
from  the  bottom. 

5.  Which  color  is  third  from  the  top  ?     Second 
from  the  top  ?     Third  from  the  bottom  ? 

6.  To-morrow  bring  something  that  is  yellow 
and  tell  me  the  names  of  things  tfiat  you  have 
seen  that  are  yellow. 

Add  a  square  of  green. 


PRIMARY   ARITHMETIC.  41 

1.  Find  green.     Recall  objects  that  are  green. 

2.  Try  to  see  the  green  square  with  the  eyes 
closed. 

3.  Look  at  the  four  colors. 

4.  Think  of  the  four,  one  after  another,  with 
the  eyes  closed. 

Cover  the  squares. 

5.  Think  the  colors  slowly  from  the  top  down. 
From  the  bottom  up. 

6.  Name  the  colors  from  the  top  down.     From 
the  bottom  up.     Which  is  second  from  the  top  ? 
Third  from  the  bottom  ?    Second  from  the  bottom? 

7.  Which  color  do  you  like  best  ? 

Add  a  square  of  blue  and  work  in  the  same  manner  with 
the  five  as  with  the  four. 

Add  a  square  of  purple. 

Work  for  a  few  minutes  each  day  until  the  colors  can 
easily  be  seen  mentally  in  the  order  given. 

Show  a  standard  color.  Have  pupils  find  tints  and 
shades  of  this  color,  and  tell  whether  they  are  lighter  or 
darker  than  the  standard. 

Have  pupils  bring  things  that  are  shades  or  tints  of 
standard  colors. 

Using  colored  crayon  or  water-colors,  have  pupils  com- 
bine primary  colors  and  tell  whether  the  result  is  darker 
or  lighter  than  the  standard  secondary  color.  Example  : 
Mix  red  and  yellow.  Is  the  result  darker  or  lighter  than 
the  standard  orange  ? 

Why  is  it  one  of  the  first  duties  of  the  schools  to  test 
the  senses  and  to  devise  means  for  their  development  ? 


42  PRIMARY    ARITHMETIC. 

Handling  solids.  —  Cover  the  eyes. 


Have  a  pupil  handle  a  solid.     Take  it  away. 

Uncover  the  eyes.  Pupil  finds  a  solid  like  the  one 
handled. 

Cover  the  eyes. 

Give  a  pupil  a  solid.     Take  it  away.    Give  him  another. 

Are  the  solids  alike  ? 

Which  is  the  larger  ?     Which  is  the  heavier  ? 

Eepeat  the  exercise  from  day  to  day. 

Judgment  and  memory  should  be  carefully  cultivated 
through  the  sense  of  touch  as  well  as  through  the  sense 
of  sight.  Touch  and  motion  give  ideas  of  form,  distance, 
direction,  and  situation  of  bodies.  "  All  handicrafts,  and 
after  them  the  higher  processes  of  production,  have  grown 
out  of  that  manual  dexterity  in  which  the  elaboration  of 
the  motor  faculty  terminates." 

Similar  solids.  —  Have  a  pupil  select  a  solid  and  think 
of  some  object  like  it.  Have  other  pupils  guess  the  name 
of  the  object. 

Ex.  :  I  am  thinking  of  something  like  a  sphere. 

Is  it  an  orange  ? 

No,  it  is  not  an  orange. 

Is  it  a  ball  of  yarn  ? 

It  is  not. 


PRIMARY    ARITHMETIC.  43 

Relative  magnitudes.  —  Place  a  number  of  solids  on  the 
table. 

1.  Find  the  largest  solid.      Find  the  smallest 
solid. 

2.  Find  solids  that  are  larger  than  other  solids. 
Ex.  :  This  solid  is  larger  than  that  one. 

Find  solids  that  are  smaller. 

3.  Name  objects  in  the  room  larger  than  other 
objects. 

Ex.  :   That  eraser  is  larger  than  this  piece  of 
chalk. 

Name  objects  less  than  other  objects. 

4.  Give  names  of   objects   at   home   that   are 
smaller  than  other  objects. 

Ex.  :  A  cup  is  smaller  than  a  bowl. 

5.  Recall   objects  that  are  larger  than   other 
objects. 

Ex. :  An  orange  is  larger  than  a  .peach.     Some 
beetles  are  larger  than  bees. 

6.  What  animals  are  larger  than  other  animals? 

7.  Recall  objects  that  are  smaller  than  other 
objects. 

Ex.  :    A  base  ball  is   smaller   than  a  croquet 
ball. 

8.  Find   the  largest  pupil   in  the   class.     The 
smallest. 

9.  To-morrow  tell  me  the  names  of  objects  that 
are  larger  than  other  objects  and  the  names  of 
others  that  are  smaller. 


44  PRIMARY   ARITHMETIC. 

1.  Find  things  that  are  higher  than  other  things 
in  the  room. 

Ex. :  The  door  is  higher  than  that  table. 

2.  Find  the  tallest  pupil.     The  shortest. 
Compare  heights  of  pupils. 

Ex. :  Mary  is  taller  than  Harry. 
Compare  the  heights  of  other  objects. 

3.  Recall   objects  that  are  longer  than  other 
objects. 

4.  What  leaves  are  longer  than  they  are  wide  ? 
What  leaves  are  wider  than  they  are  long  ? 

5.  To-morrow  tell  me  the  names  of  other  leaves 
that  are  longer  than  they  are  wide. 

Cutting.  —  Let  the  pupils  at  first  cut  and  draw  what 
they  choose.  After  a  number  of  daily  exercises,  when  they 
have  gained  some  command  of  the  muscles,  let  them  try  to 
cut  in  outline  objects  which  you  place  before  them  or 
which  they  have  seen.  Let  the  work  be  simple. 

The  drawing  and  cutting  should  be  done  freely,  without 
the  restraint  of  definiteness.  If  you  ask  more  than  the 
pupil  can  easily  represent,  the  strained,  unnatural  tension 
interferes  with  free  muscular  action.  In  the  slow  and 
painful  effort  to  represent  perfectly,  the  mind  is  absorbed 
in  the  parts  and  is  prevented  from  seeing  the  whole. 
A  premature  demand  for  definite  action  is  a  fundamental 
error,  in  that  it  separates  thought  from  expression. 

"  The  imperative  demand  for  finish  is  ruinous  because 
it  refuses  better  things  than  finish."  — Euskin. 

"Of  course  one  cannot  understand  a  child's  picture- 
speech  at  once,  any  more  than  one  can  his  other  utterances. 
We  must  study  and  learn  it."  —  H.  Courthope  Bowen. 


PRIMARY    ARITHMETIC.  45 

Building.  —  Have  pupils  build  prisms  equal  to  other 
prisms. 

Teacher  shows  a  prism  and  the  pupils  build. 

Hold  the  attention  to  the  relative  size.  This  is  the 
mathematical  element. 

Avoid  the  analysis  of  solids  until  the  habit  of  recogniz- 
ing them  as  wholes  is  formed.  Do  not  ask  for  number  of 
surfaces,  lines,  corners,  etc.  Such  questions,  if  introduced 
prematurely,  tend  to  destroy  self-activity,  to  interfere  with 
judgments  of  relative  size  and  with  the  power  to  see 
relations. 

"  Analysis  is  dangerous  if  it  overrules  the  synthetic 
faculty.  Decomposition  becomes  deadly  when  it  surpasses 
in  strength  the  combining  and  constructive  energies  of  life, 
and  the  separate  action  of  the  powers  of  the  soul  tends  to 
mere  disintegration  and  destruction  as  soon  as  it  becomes 
impossible  to  bring  them  to  bear  as  one  undivided  force." 
—  Amiel. 

Ear  training.  —  Have  pupils  listen  and  tell  what  they 
hear. 

Have  pupils  note  sounds  when  various  objects  are  struck. 

Pupils  close  eyes.  Teacher  strike  one  of  the  objects. 
Pupils  tell  which  was  struck. 

Teacher  strike  two  or  more  objects. 

Pupils  tell  by  the  sound  the  order  in  which  they  were 
struck. 

Train  pupils  to  recognize  one  another  by  their  voices 
and  by  the  sounds  made  in  walking. 

Pupils  close  eyes  and  listen. 

Drop  a  ball  or  marble  two  feet,  then  three. 

Pupils  tell  which  time  it  fell  the  farther. 

"  There  are  two  ways,  and  can  be  only  two,  of  seeking 
and  finding  truth.  .  .  .  These  two  ways  both  begin  from 


46  PRIMARY    ARITHMETIC. 

sense  and  particulars  ;  but  their  discrepancy  is  immense. 
The  one  merely  skims  over  experience  and  particulars  in 
a  cursory  transit  ;  the  other  deals  with  them  in  a  due  and 
orderly  manner."  —  Bacon. 

"  It  appears  to  me  that  by  far  the  most  extraordinary 
parts  of  Bacon's  works  are  those  in  which,  with  extreme 
earnestness,  he  insists  upon  a  graduated  and  successive 
induction  as  opposed  to  a  hasty  transit  from  special  facts 
to  the  highest  generalizations. "  -  —  Whewell. 


Touch  and  sight  training.  —  Pupils  handle  -solids  : 

1.  Find  one  of   the   largest    surfaces  of    each 
solid. 

Ex. :  This  is  one  of  the  largest  surfaces  of  this 
solid. 

2.  Find  one  of  the  smallest  surfaces. 

3.  Find  surfaces  that  are  larger  than  other  sur- 
faces. 

Ex.  :  This  surface  is  larger  than  that  one. 

4.  Find  surfaces  that  are  smaller  than   other 
surfaces. 

5.  Compare  the  size  of  other  surfaces  in  the 
room. 


PRIMARY    ARITHMETIC.  47 

6.  Find  the  largest  surface  or  one  of  the  largest 
surfaces  in  the  room. 

7.  Close  the  eyes,  handle  solids,  and  find  largest 
and  smallest  surfaces. 

8.  Cover  the   eyes;  handle  and  tell  names  of 
blocks  and  of  other  objects. 

The  exercises  for  mental  training  are  only  suggestive  of 
many  others  which  teachers  should  devise.  Be  sure  that 
the  exercises  are  suited  to  the  learner's  mind,  and  to  his 
physical  condition. 

Visualizing.  —  Place  on  the  table  three  objects,  for 
example  :  A  box,  a  book,  and  an  ink-bottle. 


1.  What  can  you  tell  about  the  box?     About 
the  book?     About  the  ink-bottle?     Which  is  the 
heaviest?     Which  is  the  lightest?     Which  is  the 
largest  ? 

2.  Look  at  the  three  objects  carefully,  one  after 
another. 

3.  Close  your  eyes  and  picture  one  after  an- 
other. 

Cover  the  objects. 

4.  Think  the  objects  from  right  to  left.     From 
left  to  right. 


48  PRIMARY    ARITHMETIC. 

5.  Name  the  objects  from  right  to  left.     From 
left  to  right. 

6.  Which  is  the  third  from  the  right?     The 
second  from  the  left? 

"  Our  bookish  and  wordy  education  tends  to  repress  this 
valuable  gift  of  nature,  —  visualizing.  A  faculty  that  is 
of  importance  in  all  technical  and  artistic  occupations,  that 
gives  accuracy  to  our  perceptions  and  justness  to  our  gen- 
eralizations, is  starved  by  lazy  disuse,  instead  of  being 
cultivated  judiciously  in  such  a  way  as  will,  on  the  whole, 
bring  the  best  return.  I  believe  that  a  serious  study  of 
the  best  method  of  developing  and  utilising  this  faculty 
without  prejudice  to  the  practice  of  abstract  thought  in 
symbols  is  one  of  the  many  pressing  desiderata  in  the 
yet  unformed  science  of  education."-  — Francis  Galton. 

When  the  position  of  every  object  in  the  group  can 
easily  be  given  from  memory,  place  another  object  at  the 
left  or  right.  Add  not  more  than  one  object  in  an  exercise 
unless  the  work  is  very  easy  for  the  pupils. 

When  a  row  of  five  is  pictured  and  readily  named  in 
any  order,  begin  with  another  group  of  five.  Each  day 
review  the  groups  learned,  so  as  to  keep  them  vividly  in 
the  mind. 

Questions  or  directions  similar  to  the  following  will  test 
whether  the  groups  are  distinctly  seen  : 

Picture  each  group  from  the  right.  Name  objects  in 
each  from  the  right. 

In  the  third  group,  what  is  the  second  object  from  the  left  ? 

What  is  the  middle  object  in  each  group  ?  What  is  the 
largest  object  in  each  group  ? 

When  four  or  five  groups  can  be  distinctly  imaged, 
this  exercise  might  give  place  to  some  other. 


PRIMARY    ARITHMETIC.  49 

Finding  circles.  —  Show  pupils  the  base  of  a  cup,  a 
cylinder,  or  a  cone,  and  tell  them  that  it  is  a  circle. 

Conduct  the  exercises  so  that  the  doing  will  call  forth 
variety  of  expression  in  telling  what  is  done. 

The  correct  use  of  the  pronouns,  verbs,  etc.,  will  thus 
be  secured  without  waste  of  the  pupils'  energy.  What  the 
pupils  see  and  do  should  lead  to  statements  similar  to  the 
following  : 

That  circle  is  larger  than  this  one.  I  have  found 
a  circle  that  is  larger  than  that  one.  Helen  has  found 
a  circle  larger  than  that  one.  He  has  found  a  circle 
smaller  than  this  one.  They  have  found  circles  larger 
than  this  one. 

1.  Find  circles. 

2.  Find   circles   that   are   larger   than   others. 
Find  circles  that  are  smaller. 

3.  Find  the  largest  circle  in  the  room. 

4.  Find  one  of  the  smallest. 

5.  Find  circles  in  going  to  and  from  school  and 
at  home,  and  tell  me  to-morrow  where  you  saw 
them. 

Finding  forms  of  the  same  general  shape  as  those  taken 
as  types  is  of  the  highest  importance.  Unless  this  is  done 
pupils  are  not  learning  to  pass  from  the  particular  to  the 
general.  They  are  not  taught  to  see  many  things  through 
the  one,  and  the  impression  they  gain  is  that  the  particular 
forms  observed  are  the  only  forms  of  this  kind.  Unless 
that  which  the  pupil  observes  aids  him  in  interpreting 
something  else,  it  is  of  no  value  to  him.  Teaching  is 
leading  pupils  to  discover  the  unity  of  things. 


50 


PRTMAKY    ARITHMETIC. 


Finding  rectangles.  —  Show  pupils  rectangles  (faces  of 
solids),  and  tell  them  that  such  faces  are  rectangles. 

1.  Find  other  rectangles  in  the  room. 
Ex.  :  This  blackboard  is  a  rectangle. 

2.  Find   larger   and    smaller   rectangles    than 
this  one. 

3.  Find  square  rectangles.      Find  oblong  rect- 
angles. 

Finding  triangles.  —  Show  the  pupils  the  base  of  a  tri- 
angular prism  or  pyramid. 

The  base  of  this  solid  is  a  triangle. 

1.  Find  triangles  in  the  room. 

2.  Find  triangles  that  are  larger  and  smaller 
than  other  triangles. 

Finding  edges  or  lines.  —  Place  solids  where  they  can  be 
handled. 


1.    Show  edges  of  different  solids. 

Show  one  of  the  longest1  edges  of  the  largest  solid. 

1  The  form  of  the  solid  will,  of  course,  determine  the  adjective 
to  use.  Every  lesson  should  help  to  familiarize  the  child  with 
correct  forms  of  speech. 


PRIMARY    ARITHMETIC.  51 

2.  Look  for  the  longest  edges  in  each  of  the 
solids. 

3.  Show  the  longer  edges  of  other  objects  in 
the  room. 

Ex.  :  This  and  that  are  the  longer  edges  of  the 
blackboard. 

4.  Show  the  shorter  edges  of  different  objects. 

5.  Find  edges  of  different  solids  and  tell  whether 
they  are  longer  or  shorter  than  other  edges. 

Ex. :  This  edge  of  this  solid  is  shorter  than  that 
edge  of  that  one. 

6.  Find  edges  of  objects  in  the  room  and  tell 
whether  they  are  longer  or  shorter   than  other 
edges. 

Ex. :  This  edge  of  the  table  is  longer  than  that 
edge  of  the  desk. 

7.  Make  sentences  like  this  :  This  edge  is  longer 
than  that  one  and  shorter  than  this  one. 

"Vision  and  manipulation,  —  these,  in  their  countless 
indirect  and  transfigured  forms,  are  the  two  cooperating 
factors  in  all  intellectual  progress."  —  John  Fiske. 

Relative  length.  —  Scatter  sticks  of  different  lengths 
on  a  table. 

Use  one  as  a  standard.  Pupils  select  longer  and  shorter, 
and  state  what  they  have  selected. 

After  pupil  selects  a  stick  and  expresses  his  opinion, 
let  him  compare  the  sticks  by  placing  them  together. 
This  will  aid  him  in  forming  his  next  judgment. 

Select  sticks  that  are  a  little  longer  or  a  little  shorter. 
This  exercise  will  demand  finer  discrimination  than  an 


52  PRIMARY    ARITHMETIC. 

exercise  where   there  is  no  restriction  as  to  comparative 
lengths. 

Direction  and  position. — Pupils  and  teacher  point  : 

1.  Teacher:  That  is  the  ceiling.     This  is  the 
floor.     That  is  the  back  wall.     This  is  the  front 
wall.     This  is  the  right  wall.     That  is  the  left 
wall.     This  is  the  north  wall.     That  is  the  south 
wall.     This  is  the  east  wall.      That  is  the  west 
wall. 

2.  A  pupil  points  and  teacher  tells  to  what  he 
is  pointing.     A  pupil  points  and  the  pupils  tell  to 
what  he  is  pointing. 

3.  Tell  the  position  of  objects  in  the  room. 
Ex.  :  There  is  a  picture  of  a  little  girl  on  the 

north  wall.     There  are  three  windows  in  the  west 
wall. 

Place  groups  of  solids  on  three  or  four  desks  in  different 
parts  of  the  room,  thus  : 


1.    Tell  the  position  of  each. 
Ex.  :   The  cylinder  is  at  the  left  at  the  back. 
The  cube  is  at  the  right  in  front. 


PRIMARY    ARITHMETIC.  53 

2.   Without  looking  tell  where  the  objects  are. 

Tell  where  different  pupils  sit. 

Ex.  :  Mary  sits  on  the  'second  seat  in  the  fourth  row 
from  the  right. 

Place  a  number  of  objects  on  a  table. 

Let  pupils  look  not  longer  than  ten  seconds.  Cover  the 
objects.  Have  pupils  tell  what  they  saw.  Practise  until 
pupils  learn  to  recognize  objects  quickly. 

Have  a  pupil  from  another  class  walk  through  the  room. 
Ask  pupils  to  tell  what  they  observed. 

Such  exercises  as  the  following,  if  not  carried  to  the 
point  of  fatigue,  cultivate  alertness  of  mind,  concentration, 
and  power  to  respond  quickly  to  calls  for  action. 

Teacher  occupy  a  pupil's  seat,  give  directions  slowly, 
then  place  hand  where  she  wishes  the  pupils  to  place  theirs. 

1.  Place  hand  on  the  front  of  your  desk.     On 
the  back.     In  the  middle.     At  the  middle  of  the 
right  edge.     At  the  middle  of  the  left  edge.     On 
the  right  corner  in  front.     On  the  left  corner  at 
the  back.     On  the  left  corner  in  front.     On  the 
right  corner  at  the  back. 

2.  Pupil  place  hand  and  teacher  or  other  pupil 
tell  where  it  is. 

3.  Pupil  place  an  object  in  different  positions 
on  the  desk.     Pupils  tell  where  it  is. 

Give  each  pupil  a  cube.  Teacher  use  rectangular  solid 
and  follow  her  own  directions. 

4.  Place  finger  on  upper  base.     On  the  lower 
base.     On  the  right  face.     On  the  left  face.     On 
the  front  face.     On  the  back  face. 


54  PRIMARY   ARITHMETIC. 

5.  Pupils  place  finger  and  teacher  tell  where  it 
is  placed. 

6.  Pupils  place  finger  and  tell  where  they  have 
placed  it. 

Place  solids  where  they  can  be  observed. 


"We  overlook  phenomena  whose  existence  would  be 
patent  to  us  all,  had  we  only  grown  up  to  hear  it  familiarly 
recognized  in  speech."  —  William  James. 

1.  Tell  the  names  of  as  many  as  you  can. 

2.  What  is  the  name  of  the  first  at  the  left? 
Give  name  if  none  of  the  pupils  know  it.     Of  the 
second  ?     Of  the  third  ?     Of  the  first,  second,  and 
third  ?    Of  the  fourth  ?    Of  the  first,  second,  third, 
fourth?     Of  the  fifth  ?     Of  the  five  ? 

3.  Look  at  the  solids.      Then  think  of  them 
without  looking. 

Cover  the  solids. 

4.  Give   names   in   order   from   left   to  right. 
From  right  to  left. 

5.  Tell  position. 

Ex. :  The  square  prism  is  the  second  solid  from 
the  left. 


PRIMARY    ARITHMETIC.  55 

Building.  —  Give  pupils  a  number  of  cubic  inches. 

1.  Build  a  prism  equal  to  this  one  (show  prism 
only  for  an  instant). 

Build  a  prism  equal  to  this  one. 

Build  a  cube  equal  to  this  one. 

Give  other  similar  exercises  from  day  to  day. 

Cutting.  — 1.   Cut  a  slip.     Cut  a  longer  slip. 

2.  Cut  a  slip.     Cut  a  shorter  slip. 

Give  each  pupil  a  square  two  inches  long. 

3.  Cut   larger   squares    than   the    square   two 
inches  long. 

What  did  I  ask  you  to  cut  ? 

4.  Cut  smaller  squares   than   the   square   two 
inches  long. 

What  did  I  ask  you  to  cut  ? 

5.  Cut   a   square    that   is   neither   larger   nor 
smaller  than  the  square  two  inches  long. 

Give  other  exercises. 

"  Almost  invariably  children  show  a  strong  tendency  to 
cut  out  things  in  paper,  to  make,  to  build,  —  a  propensity 
which,  if  duly  encouraged  and  directed,  will  not  only  pre- 
pare the  way  for  scientific  conceptions,  but  will  develop 
those  powers  *of  manipulation  in  which  most  people  are 
most  deficient." — Herbert  Spencer. 

Drawing.  —  1.  Draw  a  square.  Draw  a  smaller 
square. 

2.  Draw  a  large  square,  a  small  square,  and 
one  larger  than  the  small  square  and  smaller 
than  the  large  square. 


56  PRIMARY    ARITHMETIC. 

3.  Draw  two  equal  squares. 

4.  Draw  a  line.     Draw  a  longer  line. 

5.  Draw  a  line.     Draw  a  shorter  line. 

6.  Draw  a  line.     Draw  another  neither  longer 
nor  shorter  than  this  line.    Draw  other  equal  lines. 

Do  not  push  demands  in  advance  of  the  child's  growing 
power  to  do. 

Through  the  child's  attempts  to  do  that  which  it  wishes, 
comes  the  fitting  of  the  muscles  for  more  definite  and  more 
complex  movements.  Above  all  things  let  the  earlier 
movements  be  pleasurable,  that  an  impulse  to  renewed  exer- 
tion may  be  given.  The  desire  to  create  is  the  truest 
stimulus  to  that  action  which  gives  muscular  control.  Our 
exactions  may  make  the  doing  so  disagreeable  as  to  destroy 
the  desire  to  produce. 

Relative  magnitude.  —  Place  solids  where  they  can  be 
handled. 

1.  Find  solids  that  are  a  little  larger  than  other 
solids. 

2.  Find  solids  that  are  a  little  smaller. 

3.  Find  objects  that  are  a  little  larger  or  a 
little  smaller  than  other  objects. 

Ex. :  That  desk  is  a  little  larger  than  this. 

4.  Find  surfaces  of  the  solids  that  are  a  little 
larger  or  a  little  smaller  than  other  surfaces. 

5.  Find  edges  of  the  solids  that  are  a  little 
longer  or  a  little  shorter  than  other  edges. 

6.  Find  edges  of  other  objects  that  are  a  little 
longer  and  those  that  are   a  little  shorter  than 
other  edges. 


PRIMARY   ARITHMETIC.  57 

Cutting.  —  1.  Cut  a  slip  of  paper.  Cut  another 
a  little  longer.  Another  a  little  shorter.  Measure. 
Practise. 

2.  Cut  a  square.  Cut  another  a  little  larger. 
Another  a  little  smaller.  Measure.  Practise. 

Drawing.  —  1.  Draw  a  line.  Draw  another  a 
little  longer.  Another  a  little  shorter.  Measure. 
Practise. 

2.  Draw  a  square.  Draw  another  a  little  longer. 
Another  a  little  smaller.  Measure.  Practise. 

Cutting.  —  1.  Cut  a  slip  of  paper.  Try  to  cut 
another  equal  in  length  to  the  first.  Look  at 
them.  Which  is  the  longer  ?  Place  them  together 
to  see  if  they  are  equal.  Practise  cutting  and 
comparing. 

Give  each  pupil  paper  and  an  oblong  rectangle. 

2.  Cut  a  rectangle  as  large  as,  or  equal  to,  the 
rectangle  I  have  given  you.  What  are  you  to 
cut  ?  Is  the  rectangle  you  cut  as  long  as  the 
rectangle  I  gave  you  ?  Is  it  as  wide  ?  Does  the 
one  you  cut  exactly  cover  the  one  I  gave  you  ? 
Are  the  two  rectangles  equal  ?  Practise  trying  to 
cut  a  rectangle  exactly  the  same  size  as  or  equal 
to  the  one  I  gave  you. 

Equality.  —  "  The  intuition  underlying  all  quantitative 
reasoning  is  that  of  the  equality  of  two  magnitudes."  — 
Herbert  Spencer. 

1.   Find  solids  and  other  objects  that  are  equal. 


58  PRIMARY    ARITHMETIC. 

2.  Find   solids   in  which   the  surfaces  are   all 
equal. 

3.  Find  solids  that  have  surfaces  of  only  two 
sizes. 

4.  Find  solids  that  have  surfaces  of  three  sizes. 

5.  Find  solids  in  which  the  edges  are  all  equal. 

6.  Find  solids  that  have  edges  of  two  different 
lengths. 

7.  Find  solids  that  have  edges  of  three  different 
lengths. 

8.  Find  a  solid  that  has  four  equal  surfaces. 
How  many  other  equal  surfaces  has  it? 

9.  Find  a  solid  that  has  two  equal  large  sur- 
faces. 

10.  Find  a  solid  that  has  two  equal  small  sur- 
faces. 

11.  Find  a  solid  that  has  four  equal  long  edges. 

12.  Show  me  an  edge  of  one  solid  equal  to  an 
edge  of  another. 

13.  Show  me  two  edges  of  a  solid  which,  if  put 
together,  will  equal  one  edge  of  another. 

14.  Find  objects  in  the  room  that  are  equal,  or 
of  the  same  size. 

Ex. :  Those  two  windows  are  equal.    Those  two 
erasers  are  equal. 

Give  each  pupil  a  square. 

1.    Cut  a  square  equal  to  the  one  I  have  given 
you.    Compare.     Is  the  square  you  have  cut  equal 


PBIMAKY    AK1THMETIC.  59 

to  the  one  I  gave  you  ?    Practise  cutting  and  com- 
paring. 

Give  each  pupil  a  triangle. 

2.  Cut  a  triangle  equal  to  the  one  I  have  given 
you?  Compare.  Are  they  equal?  Which  is  the 
larger  ? 

1.  Draw  a  line.     Draw  another  equal  to  the 
first.     Measure.     Are  the  lines  equal  ? 

Give  each  pupil  a  square. 

2.  Draw  a  square  equal  to  the  one  I  have  given 
you.     Do  the  squares  look  exactly  alike  ?     Meas- 
ure.    Are  they  equal  ? 

3.  Draw  a  triangle.     Draw  an  equal  triangle. 
Do  the  triangles  look  exactly  alike?     Are  they 
equal  ? 

1.  Show  me  equal  surfaces  in  the  room.     Equal 
edges. 

2.  Show  me  the  equal  long  edges  of  the  black- 
board.     How   many   equal   long   edges    has   the 
blackboard?     How  many  short?     Show  me  the 
two  equal  long  edges  and  the  two  equal  short 
edges  of  other  surfaces. 

3.  Show  me  the  two  largest  surfaces  of  this 
box. 

4.  A  chalk-box  has  surfaces  of  how  many  sizes? 
Show  a  real  brick  or  a  paper  model. 

5.  How  many  equal  large  surfaces  has  a  brick  ? 


60  PEIMARY  ARITHMETIC. 

How   many   equal   small   surfaces?     How  many 
other  equal  surfaces  ? 

6.  Show  me  a  surface  in  one  solid  equal  to  a 
surface  in  another. 

7.  Show  me  two  surfaces  which,  if  put  together, 
will  equal  one  surface  that  you  see. 

8.  Show  me  one  of  the  longest  edges  of  this 
box.     One   of   the    shortest.     One   of   the   other 
edges. 

9.  How  many  equal  long  edges  has  the  bc5x? 
How  many  equal  short  edges  ?     How  many  other 
equal  edges  ? 

10.  How  many  rows  of  desks  do  you  see  ? 

11.  Show  me  two  equal  rows. 

Pupil  observe  objects.  Cover  his  eyes.  Let  another 
pupil  substitute  an  object  for  one  of  those  observed. 
Uncover  eyes.  Pupil  tell  what  was  taken  away  and  what 
was  put  in  its  place. 

Secure  sets  of  squares  and  of  other  rectangles  of  differ- 
ent dimensions.  Scatter  sets  over  the  table.1 

Train  pupils  to  select  those  that  are  equal. 

Ex.  :  That  square  rectangle  equals  this  one,  or  that 
oblong  rectangle  equals  this  one,  or  James  found  a  square 
equal  to  this  one. 

Secure  variety  of  statement. 

Cutting. —  1.  Look  at  a  cube  2  in.  long  and  cut 
a  square  equal  to  one  of  its  surfaces,  or  look  at  a 

1  Length  of  squares,  —  2  in.,  2|  in.,  3  in.,  3 £  in.,  4  in.  Dimen- 
sions of  oblong  rectangles,  —  1  X  2,  2x2,  3x2,  4x2,  5x2, 
and  others  1  X  3,  2  x  3,  3  X  3,  4  X  3,  5  X  3,  6  X  3. 


PRIMARY    ARITHMETIC.  61 

square  rectangle  2  in.  long  and  cut  an  equal  one. 
What  did  I  ask  you  to  cut  ? 

Let  pupils  criticise  their  own  work.  Do  not  tell  them 
that  the  square  rectangle  they  cut  is  too  large  or  too  small  ; 
let  them  compare  and  tell  you.  The  work  will  be  good, 
no  matter  how  crude  or  imperfect,  if  it  is  the  best  the 
pupil  can  do.  Growth  is  possible  only  from  the  basis  of 
genuine,  natural  expression. 

2.  Practise  cutting  and  comparing. 

3.  Cut  a  square  rectangle  two  inches  long  with- 
out observing  model. 

4.  Cut  a  rectangle  whose  length  and  width  are 
the  same.     Measure.     Are  they  equal  ?     What  is 
the  name  of  this  figure  ?     Practise. 

To-morrow,  have  pupils  cut  the  square  rectangle  again. 

Have  them  tell  what  they  cut,  in  order  to  learn  to  asso- 
ciate the  language  with  the  thing. 

Give  pupils  square  rectangles  four  inches  long  and  train 
them  to  cut,  first  when  observing,  then  from  memory. 

Give  pupils  rectangles  4  in.  by  2  in.,  and  tell  them  to 
cut  rectangles  4  in.  by  2  in. 

5.  What  did  I  tell  you  to  cut  ?     After  cutting, 
compare  and  measure. 

6.  What  are  the  names  of  the  three  forms  that 
you  have  cut  ? 

7.  What  is  the  width  of  the  square  2  in.  long  ? 
Of  the  square  4  in.  long  ? 

Why  are  a  child's  ideas  necessarily  crude  rather  than 
complete  ?  What,  then,  should  be  true  of  his  outward 
representations  ? 


OF  THB 

UNIVERSITY 


62  PRIMARY   ARITHMETIC. 

Why  is  it  impossible  to  secure  perfect  forms  from 
young  children  without  interfering  with  mental  and  moral 
development  ? 

"  We  shall  not  begin  with  a  pedantic  and  tiresome  insist- 
ence on  accuracy  (which  is  not  a  characteristic  of  the 
young  mind),  but  endeavor  steadily  to  lead  up  to  it  —  to 
grow  it  —  producing  at  the  same  time  an  ever-increasing 
appreciation  of  its  value." — H.  Courthope  Bowen. 

As  before  urged,  let  the  work  be  done  freely.  Unnatural 
restraint  in  expressing  results  in  lack  of  feeling.  It 
lessens  desire  to  see  and  to  do.  The  use  of  things  in 
which  mathematical  relations  are  conspicuous  furnishes 
no  excuse  for  disregarding  the  truth  that  progress  in  the 
power  to  represent  either  within  or  without  is  ever  from 
the  less  to  the  more  definite.  The  child  is  not  troubled 
by  a  complexity  or  a  definiteness  which  it  does  not  see. 
Teaching  in  harmony  with  nature  will  permit  the  child  to 
see  freely  and  express  freely. 

Exercise  in  judging  will  gradually  increase  the  power 
of  definite  thinking  ;  and  exercise  in  doing  the  power  of 
definite  action. 

Drawing.  —  Draw  6-in.  squares  on  different  parts  of  the 
blackboard. 

Pupils  observe  and  try  to  draw  equal  squares.  Meas- 
ure, and  try  again. 

Let  one  pupil  draw  and  others  estimate  whether  the 
square  is  larger,  smaller,  or  equal  to  the  6-in.  square. 

Have  pupils  measure  after  drawing,  so  that  they  may 
see  mistakes  and  make  more  accurate  estimates.  v 

Draw  lines  a  foot  long.  Pupils  observe  the  lines  and 
try  to  draw  equal  lines. 

Let  one  pupil  draw  and  others  estimate  whether  the 
lines  are  longer,  shorter,  or  equal. 


PRIMARY    ARITHMETIC.  63 

Pupils  find  edges  of  objects  that  they  think  are  a  foot 
long. 

Without  pupils  observing  you,  draw  lines  a  foot  long, 
a  little  more  than  a  foot  long,  and  others  a  little  less  than 
a  foot  long. 

Arrange  obliquely,  horizontally,  and  vertically.  Letter 
A,  B,  (7,  etc. 

Pupils  select  different  lines.  Ex.  :  The  line  C  is  less 
than  a  foot  long.  Other  pupils  tell  whether  they  agree 
or  not. 

Have  pupils  find  edges  in  the  room  a  little  more  or  a 
little  less  than  a  foot  long. 

Without  pupils  observing  you,  draw  a  line  2  ft. 
long. 

Have  pupils  estimate  the  length.     Let  them  measure. 

Without  pupils  observing  you,  draw  lines  on  the  board 
less  than  2  ft.,  more  than  2  ft.,  and  2  ft.  Letter. 

Have  pupils  estimate  the  lengths.  Ex.  :  I  think  the 
line  B  is  more  than  2  ft.  long.  Measure. 

Have  pupils  find  edges  in  room  a  little  more  or  a  little 
less  than  2  ft.  long. 

Draw  a  6-in.  line  on  the  board.  Do  not  separate  into 
inches.  Draw  a  foot.  Pupils  look  at  both  lines.  How 
many  6-in.  lines  in  the  foot  ? 

Draw  a  4-in.  line.  Pupils  observe  and  draw.  Observe 
the  foot  and  the  4-in,  line.  How  many  4-in.  lines  in  a 
foot? 

Place  the  solids  where  they  can  be  handled.  Pupils 
estimate  the  length  of  edges.  Measure. 

Have  pupils  show  edges  of  solids  that  they  think  are 
4  in.  long. 

Have  pupils  tell  how  long,  wide,  and  high  they  think 
each  solid  is.  Ex.  :  I  think  this  solid  is  4  in.  long,  2  in. 
wide,  and  1  in.  high,  or  it  is  4  in.  by  2  in.  by  1  in. 


64  PBIMARY    ARITHMETIC. 

"  If  the  judgment  made  be  original,  then  the  standpoint 
of  the  one  making  the  judgment  is  disclosed."  —  William 
T.  Harris. 

Building.  —  If  a  direction  is  not  understood,  the  teacher 
should  explain  by  doing  a  thing  similar  to  that  she  wishes 
done.  Thus,  if  she  says  build  a  unit  equal  to  f  of  this 
one,  and  the  pupils  do  not  understand,  she  should  build  a 
unit  equal  to  f  of  it.  Then  the  pupils  should  build  units 
equal  to  f  of  other  units. 

1.  Using  cubes,  make  a  prism  equal  to  this  one. 

2.  Using  cubes,  make  a   prism  two  times  as 
large  as  this  one. 

Continue  to  build  prisms  two  times  as  large  as  those 
selected  until  this  can  be  done  easily. 

3.  Build  a  block  equal  to  f  of  this  one. 

4.  Build  one  equal  to  f  of  this  one.     Of  this 
one. 

5.  Build  a  block  equal  to  ^  of  this  one.     ^  of 
this  one. 

"Doing,  or  rather,  expressive  doing,  reveals  to  the 
teacher  the  nature  of  his  pupil's  knowledge  ;  exhibits  to 
the  pupil  new  connections  and  suggests  others  still ;  de- 
velops skill  or  effectiveness  in  doing  as  mere  exercise  of 
information  seldom  does,  or  does  but  feebly ;  and  trains 
the  muscles,  the  nerves,  and  the  organs  of  sense  to  be 
willing,  obedient,  effective  servants  of  the  mind."  —  H. 
Courthope  Bowen. 

Cutting.  —  Give  pupils  paper  rectangles  of  different 
sizes. 


PRIMARY    ARITHMETIC.  65 

1.  Cut  a  rectangle  into  two  equal  parts.     After 
cutting,  place  the  parts  together  to  see  if  they  are 
equal.     Practise  cutting  and  comparing  the  two 
parts. 

2.  Cut  rectangles  into  three  equal  parts.     Com- 
pare the  parts.     Are  they  all  equal  ?     Practise. 

Drawing.  —  1.  Draw  a  line.  Place  a  point  in 
the  middle  of  the  line.  Measure  to  see  if  the 
parts  are  equal.  Try  again.  Measure.  Is  one  of 
the  parts  longer  than  the  other  ?  Are  the  parts 
equal  ?  What  is  meant  by  equal  ?  Show  me  one 
of  the  two  equal  parts.  Show  me  the  other. 

2.  Draw  a  line.      Separate  it  into  two  equal 
parts.     Measure.     Are  the  parts  equal  ?    Separate 
the  line  into  four  equal  parts.     Show  me  one  of 
the  four  equal  parts.     Show  me  three  of  the  four 
equal  parts.     Show  me  the  four  equal  parts. 

3.  Draw  a  line.     Separate  it  into  three  equal 
parts.     Measure.     Are  the  parts  equal  ? 

4.  Show  me  where  the  line  should  be  drawn  to 
separate    the   blackboard   into   two   equal    parts. 
Point  to  the  two  equal  parts  of  the  board. 

5.  Can  you  see  the  two  equal  parts  of  the  floor? 
Of  the  top  of  your  desk?      Show  me  two  equal 
parts  of  other  things  in  the  room. 

Give  each  pupil  a  square. 

6.  Measure  the  edges  of  the  square.     What  is 
true    of   the    edges   of   the    square?     Find   other 
squares  in  the  room. 


66  PRIMARY   ARITHMETIC. 

7.  Draw  a  square.     Measure.     Are  the  edges 
equal  ?      How  many  equal  edges   has  a  square  ? 
Practise  trying  to  draw  squares. 

8.  Draw   an   oblong   rectangle.      Measure   the 
two  long  edges.     Are  they  equal?     Measure  the 
two  short  edges.     Are  they  equal  ?     Practise  try- 
ing to  draw  oblong  rectangles. 

Equality.  — Place  solids  where  they  can  be  handled. 

1.  Show  a  part  of  that  solid  equal  to  this  one. 

2.  Show  a  part  of  one  solid  equal  to  another. 

3.  Show  a  part  of  that  rectangle  equal  to  this 
one. 

4.  Show  other  parts  that  are  equal. 

5.  What  part  of  that  solid  equals  this  one  ? 
(Give  the  name  of  the  part  if  none  of  the  pupils 
know  it.) 

6.  Show  the  part  of  that  rectangle  equal  to 
this  one. 

7.  What  is  the  name  of  the  part  of  that  rect- 
angle equal  to  this  one  ? 

Building.  —  Give  pupils  cubes.     Show  a  unit. 

1.  Build  a  unit  equal  to  this  one. 

2.  Separate  the  unit  into  two  equal  parts. 

3.  This  is  J  of  the  unit. 

Show  the  other  half.     Hold  up  the  f . 
Put  the  halves  together.     Put  one  half  on  the 
top  of  the  other. 


PRIMARY   ARITHMETIC.  67 

Show  a  larger  unit. 

4.  Build  a  unit  equal  to  ^  of  this  one. 

5.  Build  another  unit  equal  to  f  of  ito 

6.  Build  another  unit  equal  to  f  of  it. 

Relative  Magnitude.— 1.  Draw  a  line.  Sepa- 
rate it  into  two  equal  parts.  This  is  ^  of  the 
line.  Show  me  the  other  half.  Show  me  the  f 
of  the  line. 

2.  Show  me  J  of  the  top  of  your  desk.     Show 
me  £  of  the  blackboard.     Show  me  %  of  this  solid. 
Show  me  ^  of  that  solid.     Show  me  f  of  that 
solid. 

3.  Draw  a  line.     Draw  another  as  long  as  J  of 
the  first.     Measure. 

4.  Draw  a  line.     Draw  another  two  times  as 
long.     Show  me  the  part  of  the  second  line  that 
is  as  long  as  the  first.     What  part  of  the  second 
line  equals  the  first?     The  first  line  is  as  long  as 
what  part  of  the  second  ?     The  first  line  equals 
what  part  of  the  second  ? 

5.  Cut  a  slip  of  paper.     Cut  another  slip  J  as 
long.     Measure.     Cut  a  slip  of  paper.      Cut  an- 
other equal  to  J  of  the  first.     What  did  I  ask  you 
to  do? 

6.  Cut  a  rectangle.     Cut  another  two  times  as 
large.     Show  me  the  second  rectangle  you  cut. 
What  part  of  the  second  rectangle  is  as  large  as 
the  first  ? 


68  PRIMARY    ARITHMETIC. 

7.  Use  sticks  and  lay  lines  two  times  as  long  as 
other  lines. 

8.  Use  sticks  and  make  rectangles  two  times  as 
large  as  other  rectangles. 

Have  pupils  handle  solids  and  tell  into  how  many  equal 
smaller  solids  a  larger  solid  can  be  cut. 

Avoid  the  frequent  use  of  any  particular  solid,  surface, 
or  line,  in  making  comparisons.  To  use  an  inch  cube,  a 
two-inch  cube,  a  foot,  or  a  yard  in  the  elementary  work 
oftener  than  other  units  are  used  interferes  with  free 
mental  action. 

Place  on  the  table  various  solids,  cardboard  rectangles, 
both  square  and  oblong,  and  other  objects.  Let  each 
pupil  take  one  object. 

Teacher :  John,  what  have  you  ? 
I  have  a  sphere. 

Other  pupils  tell  what  they  have.  Pupils  tell  what 
other  pupils  have.  Ex.  :  William  has  a  red  square. 

Teacher:  Who  has  the  largest  solid?  Who  have  solids 
that  are  alike  ? 

Place  objects  upon  other  objects  and  tell  what  was  done. 

Ex.  :  I  put  a  cone  upon  a  cube.  Mary  placed  a  cone 
upon  a  cube. 

Place  two  objects  together  and  tell  what  you  did. 

Ex.  :  I  put  a  square  and  an  oblong  rectangle  upon  the 
table. 

Tell  what  are  in  a  group  of  three  objects. 

Ex. :  A  knife,  a  pen,  and  a  pencil  are  in  that  group. 
I  have  a  sphere,  a  prism,  and  a  cylinder. 


PRIMARY    ARITHMETIC. 


Relative  magnitude.  —  1.   Tell  all  you  can  about 
A  and  B. 


2.  B  is  as  large  as  how  many  As  ? 

3.  What  part  of  B  is  as  large  as  At     A  equals 
what  part  of  B  ? 

4.  B  equals  how  many  times  A  ? 

Place  pairs  of  solids  having  the  ratio  two  where  they 
can  be  handled  by  the  class. 

5.  Observe  solids  and  make  sentences  like  this  : 
This  solid  can  be  cut  into  two  solids  each  as  large 
as  that  one. 

6 .  Have  pupils  discover  all  the  relations  they  can. 

The  things  between  which  the  relation  ^-,  -J-,  f ,  J,  2,  3,  4, 
etc.,  is  seen,  should  vary.  Keep  in  view  the  fact  that  the 
thing  is  its  relations.  (See  page  19.)  That  which  the  pupil 
sees  as  ^  when  related  to  a  unit  twice  its  size  he  should 
see  as  -J-  or  2  according  to  that  with  which  it  is  compared. 
He  will  do  so  if  there  is  a  proper  presentation.  At  first 
his  perceptions  of  these  relations  will  be  dim.  They  will 
gradually  develop  according  to  his  experience. 

"  There  must  be  accumulation  of  experiences,  more 
numerous,  more  varied,  more  heterogeneous  —  there  must 
be  a  correlative  gradual  increase  of  organized  faculty."  — 
Herbert  Spencer. 


70 


PRIMARY    ARITHMETIC. 


"The  formation  of  an  idea  is  anorganic  evolution  which 
is  gradually  completed,  in  consequence  of  successive  expe- 
riences of  a  like  kind."  —  Dr.  Maudsley. 


c 

a 

Draw  the  units  on  the  blackboard,  making  C  6  in.  long. 

1.  Tell  all  you  can  about  these  units  or  rect- 
angles.   How  many  of  these  rectangles  are  square? 
How  many  oblong  ? 

2.  Find  the  units  that  are  equal. 

3.  The  different  units  can  be  cut  into  what? 
Ex. :  The  unit  Tcan  be  cut  into  two  M's. 

4.  Make  sentences  like  this  :    One  half  of  B 
equals  C. 

5.  Find  units  equal  to  ^  of  other  units. 

6.  Find   units   two    times   as    large    as    other 
units. 


PRIMARY   ARITHMETIC. 


71 


7.   Draw  the  units  again  to  a  different  scale  and 
continue  the  work. 


Draw  the  units  on  the  blackboard,  making  B  6  in.  long. 

1.  Find  out  all  you  can  about  these  units. 

2.  Find  the  equal  units. 

3.  Find  units  equal  to  ^  of  other  units. 
Ex. :  The  unit  /  equals  ^  of  0. 

4.  Make  sentences  like  this  :    One  half  of  M 
equals  A. 

5.  The  different  units  can  be  made  into  what 
units  ? 

Ex.  :  The  unit  M  can  be  made  into  four  (7's. 

6.  Find  units  two  times  as  large  as  other  units. 
Ex.  :  The  unit  M  equals  two  times  A. 

The  number  of  repetitions  needed  will  depend  greatly 
upon  the  manner  of  presentation.  But  no  art,  no  mode  of 
work  can  alter  the  fact  that  time  is  required,  that  ideas 
are  the  result  of  an  organizing  process. 

Building.  — Show  a  prism  3  by  1  by  1. 
1.    Build  a  unit  equal  to  this  one. 


72  PRIMARY    ARITHMETIC. 

2.  Separate  the  unit  into  three  equal  parts. 

3.  Show  me  the  three  equal  parts. 

4.  Hold  up  two  of  the  three  equal  parts. 

5.  Show  me  one  of  the  three  equal  parts. 
Show  a  unit  6  by  1  by  1. 

1.  Build  a  unit  equal  to  this  one. 

2.  Separate  the  unit  into  three  equal  parts. 

3.  Show  me  the  three  equal  parts. 

4.  Show  me  one  of  the  three  equal  parts. 

5.  Show  me  two  of  the  three  equal  parts. 
Show  a  unit  3  by  2  by  1. 

1.  Build  a  unit  equal  to  one  of  the  three  equal 
parts. 

2.  Build   another   equal  to  two    of   the   three 
equal  parts. 

3.  Build  another  equal  to  the  three  equal  parts. 

Cutting.  —  Give  each  pupil  several  rectangles  of  differ- 
ent sizes. 

1.  Cut    a    rectangle    into    three    equal    parts. 
What  did  I  tell  you  to  do  ?     Place  the  three  parts 
together.     Are  the  three  parts  equal  ?      Practise 
cutting  and  comparing. 

Drawing. —  1.  Draw  a  line.  Separate  it  into 
three  equal  parts.  Measure.  Is  one  of  the  parts 
shorter  than  one  of  the  others? 

2.  Draw  lines  of  different  lengths  and  practise 
trying  to  divide  them  into  three  equal  parts. 


PRIMARY  ARITHMETIC.  73 

3.  Draw  rectangles  of  different  sizes  and  prao 
tise  trying  to  separate  them  into  three  equal  parts* 

4.  Show  me  where  lines  should  be  drawn  to 
separate   the  blackboard   into    three  equal  parts. 
Move   your  hand  over  each  of    the   three  equal 
parts  of  the  blackboard. 

Select  different  solids. 

5.  Show  me  where  each  should  be  cut  to  sepa- 
rate it  into  three  equal  parts. 

6.  Find  a  solid  that  can  be  made  into  three 
parts,  each  as  large  as  this  solid. 

Ex.:  That  solid  can  be  made  into  three  solids 
each  as  large  as  this  one. 

Give  each   pupil   a  piece  of  paper  on  which  there  is 
drawn  a  line  equal  to  D. 


1.  Draw  a  line  equal  to  D. 

2.  Draw  a  line  two  times  as  long  as  D. 

3.  Draw  a  line  three  times  as  long  as  D. 

4.  Name  the  lines  D,  A,  B. 

5.  A  is  how  many  times  as  long  as  D  ? 

6.  B  is  how  many  times  as  long  as  Z)? 

7.  Show  me  ^  of  A.     B  is  how  many  times  as 
long  as  i  of  A  ? 

8.  Show  me  ^  of  A.     Draw  a  line  three  times 
as  long  as  J  of  A. 

9.  Draw  a  line  equal  to  the  sum  of  D  and  A. 


74  PRIMARY   ARITHMETIC. 

The  sum  of  D  and  A  equals  what  line  ? 

10.  If  we  call  D  I,  what  ought  we  to  call  A  ? 
What  ought  we  to  call  B  ? 

11.  The  sum  of  A  and  D  equals  what?     The 
sum  of  1  and  2  equals  what  ? 

Relative  magnitude.  —  Give  each  pupil  a  square  inch 
and  an  oblong  2  in.  by  1  in.  and  another  3  in.  by  1  in. 


1.  What  is  the  length  of  the  square  rectangle  ? 
How  long  is  the  largest  rectangle  ?     What  is  the 
length  of  the  other  rectangle  ? 

2.  Show  me  the  rectangle  2  in.  by  1  in.     The 
rectangle  3  in.  by  1  in.     Point  to  each  rectangle 
and  describe  it. 

Ex.  :  This  is  a  square  rectangle  1  in.  long. 

3.  Call  the  largest  rectangle  B9  the  smallest  0, 
and  the  other  N.     Show  me   0.     Show  me  B. 
Show  me  N. 

4.  JVis  as  large  as  how  many  O's  ?    What  part 
of  N  equals  0  ?    N  equals  how  many  times  0  ? 
0  equals  what  part  of  JV? 

5.  B  is  as  large  as  how  many  O's  ?     B  equals 
how  many  times  0  ?    Show  me  £  of  N.    B  is  how 
many  times  as  large  as  J  of  JV? 

6.  If  we  call  0  £,  what  is  Nt     What  is 

7.  Cut  rectangles  equal  to  0,  JV,  and  B. 


PRIMARY    ARITHMETIC. 


75 


1.    Place  0  and  ^together  and  make  one  rect- 
angle of  the  two.     How  long  is  the  rectangle  you 


0 

N 

B 

have  made  ?     How  wide  is  it  ?     It  is  as  large  as 
what  rectangle  ?     It  equals  what  rectangle  ? 

2.  Place  0,  JV,  and  B  together,  making  one  rect- 
angle of  the  three.     How  long  is  the  rectangle  ? 
This  rectangle  could  be  cut  into  how  many  B's  ? 
Into  how  many  N's  ? 

3.  Show   me  ^  of  the  rectangle.     B  is  what 
part  of  the  rectangle  ?     If  you  put  two  rectangles 
together,  the   new  rectangle    is   called  the  sum. 
The  sum  of   0  and  N  is  what  part  of  the  rect- 
angle ? 

4.  If  we  call   0  1,  what  ought  we  to  call  Nt 
What  ought  we  to  call. 5?     Show  me  the  unit  3. 
The  unit  2.     The  unit  1. 

Use  different  magnitudes,  and  change  their  arrangement 
very  often.  If  this  is  not  done  the  objective  representa- 
tions become  the  thing,  and  the  relation,  which  is  the 
essence  of  the  subject,  is  not  brought  into  consciousness 
at  all.  We  prevent  the  perception  of  truth  when  our  presen- 
tation limits  the  relation  to  particular  things.  (See  preface.) 


1.  Tell  all  you  can  about 
the  units  1,  2,  and  3. 

2.  The  unit  2  is  how  many 
times  as  large  as  the  unit  1  ? 


76  PRIMARY    ARITHMETIC. 

What  part  of  2  is  as  large  as  1  ?  The  unit  3  is 
how  many  times  as  large  as  the  unit  1  ?  Show 
me  J  of  the  unit  2.  The  unit  3  is  how  many 
times  as  large  as  half  of  the  unit  2  ?  The  unit  3 
is  as  large  as  how  many  halves  of  2  ?  The  unit 
3  equals  how  many  1's? 

1.   Place  the  units  1  and  2  together.     The  sum 


1 

2 

3 

of  1  and  2  equals  what  ?     The  unit  3  is  how  much 
greater  than  the  unit  1  ? 

Ans.  :  The  unit  3  is  2  greater  than  the  unit  1, 
The  unit  3  is  how  much  greater  than  2  ?  How 
much  less  is  the  unit  2  than  the  unit  3  ?  The 
unit  1  is  how  much  less  than  the  unit  3  ?  The 
unit  3  is  as  large  as  the  sum  of  what  two  units  ? 
Two  and  what  equal  3  ?  One  and  what  equal  3  ? 

2.  Make  one  rectangle  of  1,  2,  and  3.      The 
sum  of   1  and  2  is  what  part  of  the  rectangle  ? 
The  unit  3  is  what  part  of  the  rectangle  ? 

3.  Show  me  the  two  equal  units  that  make  the 
rectangle.     Show  me  the  three  equal  units  in  the 
rectangle.     What  three  unequal  units  do  you  see 
in  the  rectangle  ?     Separate  the  rectangle  into  two 
unequal  units.     What  are  the  names  of  the  two 
unequal  units  in  the  rectangle  ? 

4.  The  rectangle  equals  how  many  3's  ?     How 
many  2's  ? 


PRIMARY    ARITHMETIC.  77 

5.  If  the  1  is  worth  a  nickel,  what  is  the  2 
worth  ? 

6.  If  you  pay  a  nickel  for  the  1,  how  many 
nickels  ought  you  to  pay  for  the  3  ? 

7.  If  2  cost  a  dime,  1  will  cost  what  part  of  a 
dime  ? 

8.  The  cost  of  the  1  equals  what  part  of  the 
cost  of  the  2  ? 

9.  The  cost  of  3  equals  how  many  times  the  cost 
of  1? 

10.  Show  me  the  part  of   3  that  will  cost  as 
much  as  2. 

11.  If  an  apple  costs  3/?  how  many  3-/  will  two 
apples  cost? 

12.  How  many  times  as  long  will  it  take  to 
walk  two  blocks  as  to  walk  one  block  ? 

13.  What  part  of  the  time  that  it  takes  to  walk 
two  blocks  will  it  take  to  walk  one  block  ? 

14.  If  three  tops  cost  6/,  what  part  of  6/  will 
two  tops  cost? 

Draw  the   three  rectangles   on   the   blackboard  to    the 
scale  of  1  foot  to  the  inch. 

1.  If  the  length  of  the  square  rectangle  is  1, 
what  is  the  length  of  each  of  the  others  ?     What 
is  the  height  of  each  ? 

2.  What  is  the  number  of  feet  in  the  length  of 
each  rectangle  ? 

3.  Show  me  the  rectangle  1  ft.  by  1  ft.     The 
rectangle  1  ft.  by  2  ft.     The  rectangle  1  ft.  by  3  ft. 


78  PRIMARY    ARITHMETIC. 

4.  Show  me  the  upper  edge  of  the  middle  rect- 
angle.    The  lower  edge.     The  right  edge.     The 
left  edge.     How  many  edges  has  each  rectangle  ? 
Show  me  the  entire  edge  or  the  perimeter  of  each. 

5.  How   many   feet   in  the   perimeter   of   the 
square  foot  ?     How  many  feet  in  the  perimeter 
of  the  middle  rectangle  ?     In  the  perimeter  of  the 
largest  rectangle  ?     Letter  the  rectangles   0,  A, 
and  B. 

6.  Have  pupils  tell  all  they  can  about  the  rela- 
tions-of  the  rectangles  0,  A,  and  B. 

7.  Name  the  rectangles   1,  2,  and   3.      Have 
pupils  tell  all  they  can  about  1,  2,  and  3.     See 
questions  on  1,  2,  and  3  in  preceding  lesson. 

Cutting 1.  Cut  a  rectangle,  making  its  length 

and  width  equal.  If  we  call  the  length  of  the 
rectangle  1,  what  ought  we  to  call  its  width? 
Practise  cutting  rectangles  whose  edges  are*l  by  1. 

2.  Cut  a  rectangle,  making  its  length  2  and  its 
width  1.  Measure.  The  length  of  the  rectangle 
is  how  many  times  its  width  ?  If  the  width  of 
this  rectangle  is  1,  what  is  its  length  ?  Cut  rect- 
angles making  the  dimensions  1  by  3.  Measure. 
Practise. 

Drawing.  —  1.  Try  to  draw  rectangles  on  the 
blackboard  1  by  1.  Measure. 

2.  Draw  rectangles  on  the  blackboard  1  by  2. 
Measure. 


PRIMARY    ARITHMETIC. 


79 


3.  Draw  rectangles  whose  edges  will  be  repre- 
sented by  1  and  2. 

4.  The  length  is  how  many  times  as  great  as 
the  height  ? 

Drawing  should  prolong  attention.  For  the  teacher, 
drawing  should  be  an  index  of  what  the  child  can  see 
and  do. 

Cutting.  —  1.  This  rectangle  is  1.  Cut  a  1,  a  2, 
and  a  3. 

2.  The  2  you  have  cut  equals   how 
many  times  the  1  ? 

3.  The  3  you   have  cut  equals  how 
many  times  the  1  ? 

4.  If  you  put  the  1,  2,  and  3  together, 

the  sum  will  make  how  many  3's  ?     How  many 
2's  ?     How  many  1's  ? 

Drawing.  —  1.    This  is  3.     Draw  a  3,  a  2,  a  1. 

2.  Have  a  pupil  draw  a  unit 
on  the  blackboard  and  name  it  1, 
2,  or  3.  Have  other  pupils  draw 
the  other  two  units.  Use  lines 
and  rectangles. 

Relative  sizes. — Place  the  cube  1 
in.  long,  the  solid  1  in.  by  1  in.  by  2  in.,  and  the  solid  1 
in.  by  1  in.  by  3  in.  where  they  can  be  seen.  Name  them 
C,  D,  and  A. 

1.  What  is  the  name  of  the  largest  unit?  The 
name  of  the  smallest  ?  Of  the  other  unit  ? 


80  PBIMAKY    ARITHMETIC. 

2.  Look  at  the  units  C,  D,  and  A,  and  tell  all 
you  can  about  them. 

3.  D  equals  how  many  C"s?     A  equals  how 
many  C  's  ?     D  is  how  many  times  as  large  as  C  ? 
A  is  how  many  times  as  large  as  C  ?    A  equals 
how  many  times  C  ? 

4.  Show  |  of  D.     What  part  of  D  equals  (7? 
A  equals  how  many  times  -|  of  D  ? 

5.  Put  C  and  D  together.     The  sum  of  C  and 
D  equals  what  unit  ?     The  sum  of  C  and  D  equals 
how  many   C's  ?      Put    C,  D,  and   A   together. 
How  many  A's  in  the  sum  ?     How  many  jD's  ? 
How  many  C's  ? 

6.  If  we  call  C  1,  what  ought  we  to  call  Z)? 
What  ought  we  to  call  A  ?    Show  me  the  1.    The 
2.     The  3. 

7.  Look  at  the  units  1,  2,  and  3,  and  tell  all 
you  can  about  them. 

8.  The  unit  2  is  as  large  as  how  many  1's  ? 
The  3  is  as  large  as  how  many  1's? 

9.  What  part  of  2  is  as  large  as  1  ?     Show  me 
the  part  of  3  that  is  as  large  as  1  ?     Show  me  the 
part  of  3  that  is  as  large  as  2. 

10.  3  is  how  much  greater  than  1  ?     3  is  how 
much  greater  than  2  ?     1  is  how  much  less  than 
3  ?     2  is  how  much  less  than  3  ?     Put  1  and  2 
together.    The  sum  of  1  and  2  equals  what  unit  ? l 

1  The  child  understands  spoken  language  before  he  uses  it ;  he 
acquires  it  unconsciously.     Let  him  have  the  same  opportunities 


PRIMARY    ARITHMETIC.  81 

Unite  the  units  1,  2,  and  3..  The  sum  equals  how 
many  3's  ?  How  many  2's  ? 

11.  If  you  put  (7,  -D,  and  A  together,  how  high 
a  post  will  they  make  ?  What  is  -|  of  the  height 
of  the  post  ?  Two  inches  equals  what  part  of  the 
height  of  the  post  ?  The  top  of  the  post  is  what 
kind  of  a  rectangle  ? 

1.  Cover  the  eyes  of  different  pupils  and  place 
solids   in   their  hands.      Let  pupils   tell  relative 
sizes  of  solids  and  surfaces  and  the  relative  lengths 
of  edges. 

2.  Find  units  that  you  can  call  1,  2,  and  3. 

3.  Show  a  unit  that  is  two  times  as  large  as 
this  one.      Show  different  units  that  equal  two 
times  other  units. 

Ex.  :  This  unit  equals  two  times  that  unit. 

4.  Show  different  units  that  equal  three  times 
other  units. 

Ex.  :  This  unit  equals  three  times  that  one. 

5.  Tell  things  like  this  :  This  is  a  2,  for  it  is  two 
times  as  large  as  that  unit. 

6.  Tell  things  like  this:  That  is  a  1?  for  this 
unit  equals  ^  of  it. 

in  learning  to  associate  ideas  with  sight-forms.  From  day  to  day 
place  upon  the  blackboard  the  expression  for  the  relations  dis- 
covered. At  first  do  not  ask  attention  to  them.  When  the  child 
wishes  to  use  them,  a  great  step  toward  the  power  to  express  will 
have  been  taken.  Let  that  which  you  write  mean  something  to 
the  child,  as  that  which  he  hears  does ;  let  it  symbolize  his  thought. 


82  PRIMARY    ARITHMETIC. 

7.  Find  solids  whose  surfaces  represent  1  and 
2.     How  many  of  the  surfaces  may  we  call  1  ? 
How  many  2  ? 

8.  Find  surfaces  of  different  solids  whose  rela- 
tions are  1,  2,  and  3. 

9.  Find  edges  that  we  may  call  1  and  2.     Tell 
how  many  1's  and  how  many  2's  you  find  in  the 
edges  of  the  solid. 

Relations  of  quart  and  pint.  —  Show  pupils  the  pint 
and  quart  measures.  Have  them  find  the  number  of  pints 
equal  to  a  quart  by  measuring. 


1.  After  measuring,  tell  all  you  can  about  the 
quart  and  the  pint. 

This  free  work  is  far  more  valuable  than  that  induced 
by  questions.  Both  the  weak  and  the  strong  have  oppor- 
tunities to  show  their  power,  while  the  exercise  tends  to 
develop  self-activity  ;  that  is,  it  fosters  a  desire  to  discover 
when  not  acting  under  the  stimulus  of  questions. 

Too  much  questioning  interferes  with  the  natural  action 
of  the  mind  in  relating  and  unifying.  It  isolates  ideas. 
It  prevents  the  teacher  from  seeing  the  real  state  of  the 
pupil's  mind.  What  is  wanted  is  a  questioning  attitude, 
—  a  curiosity  which  will  sustain  interest  and  strengthen 
attention. 


PRIMARY   ARITHMETIC.  83 

2.  What  is  sold  by  the  pint  and  by  the  quart  ? 

3.  A  quart  is  how  many  times  as  large  as  a 
pint? 

4.  What  part  of  a  quart  is  as  large,  or  as  much, 
as  a  pint  ? 

5.  A  quart  is  how  much  more  than  a  pint  ? 

6.  A  pint  is  how  much  less  than  a  quart  ? 

7.  A  quart  and  a  pint  equal  how  many  pints  ? 

8.  Show  me  1|  quarts.     What  have  you  shown 
me? 

9.  1J  quarts  equal  how  many  pints  ? 

10.  If  we  call  a  pint  1,  what  ought  we  to  call 
a  quart  ?     Why  ? 

11.  If  we  call  a  quart  2,  what  ought  we  to  call 
the  sum  of  a  quart  and  a  pint  ? 

12.  If  a  quart  is  1,  what  is  a  pint? 

Fill  the  quart  and  pint  measures  with  water,  and  let 
each  pupil  lift  the  two  measures. 

1.  Which  is  the  heavier,  —  the  quart  of  water 
or  the  pint  ? 

2.  The  quart  of  water  is  how  many  times  as 
much  as  the  pint  ? 

3.  What  part  of  the  quart  weighs  as  much  as 
the  pint  ? 

4.  The  weight  of  a  pint  equals  what  part  of 
the  weight  of  a  quart  ? 

5.  The  weight  of  a  quart  equals  the  weight  of 
how  many  pints  ? 


84  PRIMARY    ARITHMETIC. 

6.  A  pint  of  water  weighs  a  pound;  how  much 
does  a  quart  of  water  weigh  ? 

7.  What  part  of   a  quart  of  water  weighs  a 
pound  ? 

8.  The  sum  of  a  quart  and  a  pint  of  water 
weighs  how  many  pounds  ? 

9.  Compare  the  weight  of  different  solids  with 
the  weight  of  a  pint  of  water. 

Ex. :  This  solid  weighs  less  than  a  pound,  or 
this  solid  weighs  a  little  more  or  a  little  less  than 
a  pound. 

10.  If  a  pint  of  milk  costs  3/,  what  ought  a 
quart  to  cost  ? 

11.  In  a  quart  there  are  how  many  pints  ?     In 
3  quarts  there  are  how  many  2-pints  ? 

12.  How  much  milk  should  be  put  into  a  quart 
measure  to  make  it  half  full  ? 

Relation  of  the  foot  and  six  inches.  —  Have  pupils  try 
to  draw  lines  of  the  same  relative  length  as  the  foot  and 
the  6-in.   on  paper  and  on  the  black- 
board.    After  the  practice  in  drawing 
and  in  telling  what  they  can  about  the 
relations,  draw  the  foot  and  the  6-in.  lines  on  the  board. 

1.  Tell  all  you  can  about  these  lines. 

2.  What   is   the   length   of    the   longer   line? 
What  is  the  length  of  the  shorter  line  ? 

3.  Into  how  many  6-in.  can  a  foot  be  separated  ? 

4.  A  foot  is  how  much  longer  than  6  inches  ? 

5.  6  in.  and  how  many  inches  equal  a  foot  ? 


PRIMARY   ARITHMETIC.  85 

6.  6  in.  are  how  much  shorter  than  a  foot? 

7.  Show  me  the  part  of  a  foot  that  equals  6  in. 

8.  What  part  of  a  foot  equals  6  in.? 

9.  A  foot  is  how  many  times  as  long  as  6  in.  ? 

10.  6  in.  equals  what  part  of  a  foot  ? 

11.  How  many  6-in.  in  a  foot  ?     In  2  ft.  ? 

12.  Two  6-in.  equal  what? 

13.  6  in.  and  1  ft.  are  how  many  6-in.  ? 

14.  How  many  6-in.  in  1^  ft.  ? 

15.  If  we  call  6  in.  1,  what  ought  we  to  call  a 
foot? 

16.  If  a  foot  is  1,  what  is  6  in.  ? 

17.  If  we  call  6  in.  1?  what  ought  we  to  call 
2ft.? 

18.  Why  ought  we  to  call  2  ft.  4,  if  we  call 
6  in.  1  ? 

19.  Review  without  observing  the  foot  and  the 
6-in.  line. 

Relative  length.  —  Give  each  pupil  an  equilateral  tri- 
angle having  a  2-in.  base. 

1.  Cut  an  equilateral  triangle  as  large  as  this 
one.     Measure   the   edges.      Are 

they  equal  ?    Practise  cutting  and 
measuring. 

2.  Draw  an  equilateral  triangle. 
Measure    the    edges.      Are    they 
equal?     Practise  drawing  and  measuring. 

3.  Try  to  draw  a  line  equal  to  the  sum  of  two 


86  PRIMARY    ARITHMETIC. 

edges  of  the  triangle.  Is  the  line  you  have  drawn 
two  times  as  long  as  one  of  the  edges  of  the  tri- 
angle ? 

4.  Draw  a  line  equal  to  the  sum  of  the  edges  of 
the  triangle.     Is  the  line  you  have  drawn  three 
times  as  long  as  one  of  the  edges  of  the  triangle  ? 
Measure.     Try  again. 

5.  Show   me   the   perimeter   of    the    triangle. 
How  many  2-in.  in  the  perimeter  of  the  triangle  ? 

6.  Let  one  pupil  try  to  draw  an  equilateral  tri- 
angle on  the  board.     Other  pupils  criticise. 

7.  Tell  all  you  can  about  this  equilateral  tri- 
angle. 

Relation  of  the  yard  and  the  foot.  —  Draw  a  line  a 


yard  long  on  the  blackboard.     Draw  another  a  foot  long. 
Give  the  names  of  each. 

1.  What  is  the  name  of  the  longer  line?     Of 
the  shorter  line  ?     Show  me  the  yard. 

2.  Tell  all  you  can  about  the  yard  and  the  foot. 

3.  How  many  feet  do  you  think  there  are  in  a 
yard  ?    Measure.    A  yard  is  how  much  longer  than 
a  foot  ?    A  foot  is  how  much  shorter  than  a  yard  ? 

4.  A  yard  equals  howr  many  times  a  foot  ? 

5.  Into  how  many  equal  parts  must  you  sepa- 
rate a  yard  to  make  each  part  a  foot  long  ? 


PRIMARY    ARITHMETIC.  87 

6.  A  yard  of  ribbon  contains  how  many  feet  ? 

7.  Have  pupils  try  to  place  points  a  foot  apart 
on  the  blackboard.     Pupils  in  class  tell  whether 
they  are  more  or  less  than  a  foot  apart.     Measure. 
Practise. 

Estimate  the  number  of  yards  in  different 
lengths,  heights,  edges. 

Ex. :  The  height  of  that  door  is  more  than 
2  yds.  but  less  than  3. 

8.  How  many  feet  in  a  yard  ?     How  many  3-ft. 
in  2  yds.?     In  4  yds.? 

Problems.  —  1.  If  I  have  2  apples  in  my  pocket 
and  ^  as  many  in  my  hand,  how  many  have  I  in 
my  hand?  , 

2.  If  I  pay  4/  for  a  yard  of  ribbon,  how  much 
must  I  pay  for  ^  yd.  ? 

3.  If  1  ft.  of  molding  costs  2/,  how  many  2-/ 
will  1  yd.  cost  ? 

4.  If    1  ft.  of   molding  costs  17/,  how  many 
17-/  will  1  yd.  cost? 

5.  If  ^  barrel  of  flour  lasts  1  month,  how  long 
will  1  barrel  last  ? 

6.  I  use  1  yd.  of  ribbon  for  a  hat  and  f  of  a 
yard  for  a  collar ;  how  many  feet  do  I  use  ? 

7.  I  had  4  horses  and  sold  J  of  them  ;  how 
many  did  I  sell  ? 

8.  Mary  had  a  quart  of  berries  and  sold  a  pint. 
What  part  of  her  berries  did  she  sell  ? 


88  PRIMARY   ARITHMETIC. 

Relative  magnitude.  —  Show  pupils  1,  2,  3.     Call  them 
t,  t,  1- 

1.    What  are  the  names  of  these  units  ? 


2.  What  is  the  name  of  the  largest  ?     Of  the 
smallest  ?     Of  the  next  to  the  largest  ? 

3.  Put  £  and  f  together.     The  sum  of  £  and  f 
equals  what  ? 

4.  What  must  be  added  to  the  unit  f  to  make 
the  unit  1  ? 

5.  The  unit  1  is  how  much  larger  than  the  unit 

t? 

6.  You  can  separate  the  unit  1  into  how  many 
thirds  ? 

7.  What  part  of  f  is  as  large  as  J  ? 

8.  What  part  of  1  equals  the  |  ? 

9.  What  part  of  1  equals  the  f  ? 

10.  The  unit  1  is  how  many  times  as  large  as 
the*? 

11.  i  equals  what  part  of  f  ?     Of  1  ? 

12.  Show  f  of  the  top  of  this  table.     Show  f  of 
it. 


PRIMARY    ARITHMETIC.  89 

Select  other  solids  Laving  the  same  relative  size,  and 
call  them  -J-,  f ,  1.     Pupils  compare.     Tell  all  they  can. 


(Hi. 


1.  Show  f,  f,  and  |  of  different  objects  in  the 
room. 

2.  Practise  making  units  of  cubes  equal  to  f 
of  other  units. 

3.  Practise  making  units  equal  to  f  of  other 
units. 

4.  Practise  making  units  equal  to  ^  of  others. 

Give  each  pupil  a  square  inch,  a  rectangle  2  in.  by  1  in., 
and  one  3  in.  by  1  in.     Call  them  £,  f ,  1. 

1.  Tell    all   you    can    about   the    units   ^   f, 
and  1. 

2.  What  part  of  f  equals  the  J  ? 

3.  How  many  ^  in  the  1  ? 

4.  What  part  of  the  1  equals  the  £  ? 

5.  The  unit  1  is  how  many  times  as  large  as 
the  unit  £  ? 

6.  Show  me  ^   of  the  f.     The  unit  1  is  how 
many  times  as  large  as  ^  of  the  f  ? 

Draw  the  units  on  the  blackboard  to  the  scale  of  1  ft.  to 
the  inch. 


90 


PRIMARY    ARITHMETIC. 


1.  If  the  largest  unit  is  1,  what  is  the  name  of 
each  of  the  others  ? 

2.  Tell  all  you  can  about  the  relations  of  these 
units. 

Ask  questions  similar  to  those  above. 

3.  If  the  ^  is  worth  5/?  what  is  |-  worth  ? 

4.  If  the  |  is  worth  3/?  how  many  3-/  is  the  1 
worth  ? 

Draw  the  figures  of  the  diagram  on  the  blackboard; 


making  A  6  in.  long.     After  pupils  have  studied  and  com- 
pared them,  draw  to  some  other  scale. 

1.    Tell  all  you  can  about  the  relation  of  these 
units. 


PRIMARY    ARIT 


2.  If  A  is   1,  how  many  1's  in   the  diagram  ? 
Can  you  find  five  other  figures  as  large  as  A  ? 

3.  If  A  is  1,  how  many  2's  do  you  see  ?     How 
many  3's  ? 

4.  If  B  is  1,  how  many  1's  do  you  see  ?     How 
many  2's  ?     How  many  3's  ? 

5.  If   B  is    1,  how  many  of   the   figures  are 
halves  ? 

6.  If  Gr  is  1,  how  many  1's  in  the  diagram? 
How  many  2's  ?     How  many  3's  ? 

7.  If  G  is  1,  what  is  A  ?     If  G  is  1,  how  many 
of  the  figures  are  thirds  ?     How  many  represent 
|  ?     How  many  f  ?     The  figure  If  equals  how 
many  thirds? 

8.  If  A  is  a  6-in.  square,  each  of  the  others 
equals  how  many  6-in.  squares? 

9.  Make  sentences  like  this:  The  sum  of  A  and 
B  equals  G. 

Draw  a  yard,  a  foot,  and  6  in.  on  the  blackboard. 


1.  Tell  all  that  you  can  about  the  relations  of 
these  lines. 

2.  The  yard  equals  how  many  feet  ?     The  yard 
is  how  many  times  as  long  as  the  foot  ? 

3.  The  foot  is  how  many  times  as  long  as  the 


92  PRIMARY    ARITHMETIC. 

6-in.  ?      How  many  6-in.  in   the    foot  ?      In   the 
yard? 

Problems.  —  1.  The  cost  of  1  ft.  of  paving  equals 
what  part  of  the  cost  of  1  yd.  ? 

2.  1  yd.  will  cost  how  many  times  as  much  as 
i  of  a  yd.? 

3.  1  yd.  will  cost  how  many  times  as  much  as 
1  ft.? 

4.  The  cost  of  2  ft.  of   molding  equals  what 
part  of  the  cost  of  1  yd.  ? 

5.  James   has  3  marbles  and  John  has  f  as 
many  ;  how  many  has  John  ? 

6.  If  a  quart  of  milk  costs  8/,  what  part  of  8/ 
will  a  pint  cost  ? 

7.  If  a  cup  of  sugar  is  used  in  making  a  cake, 
how  many  cups  will  be  needed  in  making  a  cake 
3  times  as  large  ? 

8.  If  the  smaller  cake  is  enough  for  1  lunch, 
the  larger  is  enough  for  how  many  lunches  ? 

9.  If  3  yds.  of  tape  cost  24/,  what  part  of  24/ 
will  2  yds.  cost  ? 

10.  This  line represents  the  cost 

of  1  yd.  of  cloth  ;  draw  a  line  to  represent  the  cost 
of  |  of  a  yd. 

11.  This  line  — —  represents  the  cost  of  6  in. 
of  ribbon  ;  draw  a  line  to  represent  the  cost  of 
1  ft.     Of  1  yd. 

12.  If  $2  is  the  cost  of  ^  of  a  ton  of  coal,  what 


PKIMARY    ARITHMETIC.  93 

is  the  cost  of  1  ton  of  coal  ?     Show  relative  cost 
by  drawing  two  rectangles. 

13.  This  line represents  the  cost  of  2  ft. ; 

draw  a  line  to  represent  the  cost  of  1  yd. 

14.  2  ft.  of  cord  cost  6/.     The  cost  of  1  yd. 
equals  how  many  halves  of  6/  ? 

Draw  a  square  foot  on  the  blackboard. 

1.  Show  the  perimeter  of  the 
square    foot.      What    have    you 
shown  ?     How  many  feet  in  the 
perimeter  of  the  square  foot  ? 

2.  How    many    6-in.    lines    in 
one  edge  of  the  square  foot  ?     In 
the  perimeter  of  the  square  foot? 

Ratios  of  length.  —  Draw  a  foot  on  the  blackboard. 
Draw  a  4-in.  line.  Pupils  practise  drawing  and  meas- 
uring these  lines. 

1.  Tell  all  you  can  about  these  lines. 
Give  the  pupils  the  names  of  these  lines. 

2.  What  is  the  name  of  the  longer  line  ?     What 
is  the  name  of  the  shorter  line  ? 

3.  Into  how  many  4-in.  can  a  foot  be  divided  ? 

4.  4  in.  and  how  many  4-in.  equal  1  ft.  ? 

5.  2* 4-in.  and  how  many  inches  equal  1  ft.  ? 

6.  Show  the  part  of  a  foot  that  equals  4  in. 

7.  What  part  of  a  foot  equals  4  in.  ? 

8.  What  part  of  a  foot  equals  6  in.? 


94  PRIMARY   ARITHMETIC. 

9.    4  in.  equal  what  part  of  a  foot  ? 

10.  A  foot  is  how  many  times  as  long  as  4  in.  ? 
As  6  in.  ? 

11.  How  many  4-in.  in  a  foot  ? 

12.  Show  me  f  of  a  foot.    How  many  4-in.  in  f 
of  a  foot  ? 

13.  If  we  call  4-in.  1,  what  should  we  call  a 
foot? 

14.  If  a  foot  is  3,  what  is  4  in.  ? 

15.  If  4  in.  is  £,  a  foot  is  how  many  thirds  ? 

16.  Show  the  part  of  a  foot  that  is  2  times  as 
long  as  4  in. 

What  part  of  a  foot  is  2  times  as  long  as  4  in.  ? 
What  part  of  a  foot  equals  6  in.  ? 
Keview  without  observing  the  lines.     Have  pupils  prac- 
tise placing  dots  1  ft.  apart.     Six  inches  apart. 

Relative  size.  —  Let  pupils  handle  solids  which  repre- 
sent 1,  2,  3,  and  4.     Call  them  A,  B,  C,  and  D. 


1.    What  is  the  name  of  the  largest  solid  ?     Of 
the  smallest?     Of  the  next  to  the  largest?     Of 


PRIMARY  ARITHMETIC.  95 

the  next  to  the  smallest  ?  Give  the  names  in 
order,  beginning  with  the  smallest.  What  is  the 
name  of  the  unit  that  is  three  times  as  large 
as  At 

2.  Tell  all  you  can  about  the  units. 

Let  it  be  the  constant  practice  first  to  permit  the  pupils 
to  see  what  they  can.  The  questions  of  the  book  are  to 
aid  the  teacher  and  not  to  enslave  the  pupil.  Questions 
have  their  value,  but  when  they  force  details  upon  a  mind 
unprepared  for  them,  when  they  destroy  the  significance 
of  the  whole,  when  they  limit  individual  seeing,  when  they 
interfere  with  the  relating,  unifying  action  of  the  mind, 
they  are  intellectual  poison. 

3.  Into   how  many  A's  can  you   divide   each 
unit? 

4.  Each  unit  equals  how  many  As  ?     D  equals 
how  many  .Z?'s  ? 

5.  Place  A  and  B  together.     The  sum  of  A 
and  B  equals  what  unit  ? 

6.  Place  A  and  C  together.    The  sum  of  A  and 
C  equals  what  unit  ? 

7.  The  sum  of  A  and  C  equals  how  many  5's  ? 

8.  Place  B  and  D  together.     How  many  (7's  in 
the  sum  of  B  and  D  ? 

9.  The  sum  of  C  and  B  equals  how  many  A'a  ? 
10.    The  unit  B  is  how  many  times  as  large  as 

A  ?  The  unit  C  equals  how  many  times  A  ?  The 
unit  D  equals  how  many  times  B  ?  The  unit  D 
equals  how  many  times  A  ? 


96 


PRIMARY    ARITHMETIC. 


11.  Show  me  f  of  .C.     The  unit  D  is  how  many 
times  as  large  as  f  of  C  ? 

12.  What  part  of  D  equals  A  ?     What  part  of 
C  equals  A  ?      Show  the  part  of  D  that  is  as 
large  as  A. 

13.  Show  me  f  of  C.     What  part  of  C  is  as 
large  as  B  ? 

14.  If  A  is  2,  B  is  how  many  2's?     C  is  how 
many  2's  ?     D  is  how  many  2's  ? 


1.  Show  me  the  part  of  D  that  is  as  large  as  B. 
What  part  of  D  equals  5? 

2.  (7  is  how  many  times  as  large  as  A  ?     Show 
me  ^  of  B.     C  is  how  many  times  as  large  as  \ 
of  B? 

3.  Z)  is  how  many  times  as  large  as  A  ?     D  is 
how  many  times  as  large  as  B  ?     Show  me  J  of  (7. 
D  is  how  many  times  as  large  as  ^   of  (7?     Z) 
equals  how  many  thirds  of  C  ? 

4.  I  of   C  equals  what  unit?     }  of    C  equals 
what  part  of  B  ?     ^  of  (7  equals  J  of  what  unit  ? 


PRIMARY    ARITHMETIC.  97 

5.  Move  your  finger  from  the  top  to  the  bottom 
of  A.     Over  £  of  B.     Over  £  of  C.     Over  £  of  Z>. 
What  is  true  of  these  four  units  ?    What  units  are 
of  the  same  size  as  A  ?     Show  me  again  the  four 
equal  units.      What  are  the  names  of  the  four 
equal  units  ? 

6.  Move  your  finger  over  B.    Over  f  of  C.   Over 
^  of  D.     Show  me  the  three  equal  units  again. 
What  are  the  names  of  the  three  equal  units  ? 

7.  |  of  C  equals  what  unit?     f  of   C  equals 
what  part  of  D  ? 

8.  If  you  cut  D  into  4  equal  parts,  or  into 
fourths,  how  many  of  the  fourths  will  make  a 
unit  as  large  as  '  C  ?     f  of  D  equals  what  unit  ? 

Use  other  solids  having  the  same  relations  as  A,  B,  C, 
D.  Give  different  names  to  the  solids,  and  review.  Then 
review  without  solids. 

1.    If  we  call  A  1,  what  ought  we  to  call  5? 


2.  If  A  is  ^,  what  is  each  of  the  other  units  ? 

3.  If  B  is  f,  what  is  each  of  the  other  units  ? 

4.  If  A  is  3,  how  many  3's  in  each  of  the  other 
units  ? 

5.  If  A  is  worth  5/,  how  many  5-/  are  each  of 
the  other  units  worth? 

6.  If  A  is   a  box  which  holds   a  quart,  how 
many  quarts  will  each  of  the  other  boxes  hold  ? 
How  many  pints  will  each  box  hold  ? 


98 


PRIMARY    ARITHMETIC. 


Cutting.  —  1.  This  is  a  1.    Cut  a  1,  a  2,  a  3,  a  4. 
You  must  make  the  2  how  many  times  as  large  as 
the   1  ?      Have  you  made  the  2 
equal  to  2'1's?     Measure.     Have 
.  you  made  the  3  equal  to  3  times 
1,  or  3  times  as  large  as  the  1  ? 
Measure. 

2.   How  large  have  you  made 
the  4? 

1.  This  is  a  2.     Cut  a  1,  a  2,  a  3,  a  4.     The  1 
you  cut  equals  what  part  of  the  2  ? 

2.  The  3  you  cut  is  how  many 
times  as  large  as  ^  of  2  ?     The  4 
you  cut   is   how   many  times  as 
large  as  the  2  ? 

Drawing.  —  Let  a  pupil  draw  a  unit  on  the  blackboard, 
and  others  draw  related  units  and  tell  what  they  have 
drawn. 

Relative  size.  —  Place  solids  having  the  relation  of 
1,  2,  3,  4  where  they  can  be  handled. 

1.  If  the  smallest  unit  is  1?  what  is  the  name 
of  each  of  the  other  units?     What  is  the  name  of 
the  largest  unit  ? 

2.  Tell  all  you  can  about  the  units. 

3.  Tell  the  sums  that  you  see. 
Ex.:  The  sum  of  1  and  2  equals  3. 

4.  Tell  how  much  greater  one  unit  is  than  an- 
other.    Ex. :  4  is  three  greater  than  1. 


PRIMARY    ARITHMETIC. 


99 


5.  2  and  what  equal  4  ?     2  and  2  equal  what? 
4  equals  how  many  2's  ? 

6.  Put  4  and  2  together.     The  sum  of  4  and  2 
can  be  divided  into  how  many   3's  ?      Into  how 
many  2's  ? 

7.  The  sum  of  4  and  2  is  how  many  times  as 
large  as  3  ?     It  is  how  many  times  as  large  as  2  ? 

Give  each  pupil  a  square  rectangle  2  in.  long,  a  rectangle 
4  in.  by  2  in.,  a  rectangle  6  in.  by  2  in.,  and  a  rectangle 
8  in.  by  2  in. 


1.  Tell  all  you  can  about  the  relations  of  1,  2, 
3,4. 

2.  In  each  of  the  rectangles  2,  3?  and  4,  cover 
all  except  the  part  equal  to  1,  and  tell  what  part 
is  equal  to  the  1. 

3.  Show  all  the  parts  that  are  2  times  as  large 
as  1  and  give  the  name  of  each. 

4.  Look  for  units  that  are  equal  to  ^  of  other 
units. 

5.  Estimate  the  dimensions  of  each  of  the  rect- 
angles ;  i.e.  tell  how  long  and  wide  you  think  they 
are.     Measure.      State  the  dimensions.     Without 
observing  the  rectangles  tell   the  dimensions  of 
each. 


1.    Place  the  rectangles  1  and  2  together, 
sum  of  1  and  2  equals  what  unit  ? 


The 


100  PRIMARY    ARITHMETIC. 

2.  Show  the  part  of  the  unit  4  equal  to  the  sum 
of  1  and  2.     What  part  of  4  equals  the  sum  of  1 
and  2? 

3.  Place  the  rectangles  together  and  make  one 
rectangle  of  the  four.     Show  the  f  of  this  rect- 
angle.    The  sum  of  what  units  makes  ^  of  the 
rectangle  ?      What  units   make   the   other   half  ? 
The  sum  of  2  and  3  equals  the  sum  of  what  other 
units  ? 

4.  2  equals  what  part  of  4  ?     How  many  4's  do 
you  see  in  the  rectangle  ?      Can  you  find  2^'4's  in 
this  rectangle?      Show  me  the  2'4's.      Show  me 
the  half  of  4.     Point  to  the  2^'4's. 

5.  The  large  rectangle  can  be  made  into  how 
many  rectangles  as  large  as  3  ?     Show  me  the 
3'3's.     What  part  of  another  3  do  you  see  ? 

6.  Into  how  many  2's  can   the   rectangle   be 
made  ? 

7.  If  we  should  call  the  square  2,  what  ought 
we  to  call  each  of  the  other  rectangles  ?     If  we 
should  call  the  square  ^,  what  number  of  halves 
would  we  see  in  each  of  the  other  rectangles  ?     If 
there  are  4  sq.  in.  in  the  square  rectangle,  how 
many  4-sq.-in.  in  each  of  the  other  rectangles  ? 

Draw  the  rectangles  on  the  blackboard  to  the  scale  of 
3  in.  to  the  inch. 

8.  Tell  all  you  can  about  the  rectangles  1,  2,  3, 
and  4,  drawn  on  the  blackboard. 


PRIMARY   ARITHMETIC.  101 

Problems^ —  1.    4/  will  buy  how  many  times  as 
many  marbles  as  2/  ? 

2.  If  you  can  buy  a  barrel  of  flour  for  $4,  how 
much  can  you  buy  for  $3  ? 

3.  If  1  yd.  of  cloth  costs  $3,  how  much  cloth 
can  be  bought  for  $4  ? 

4.  If  £  of  a  basket  of  fruit  is  worth  25/?  how 
many  25-/  is  the  basket  of  fruit  worth  ? 

5.  If  1  doz.  eggs  costs  15/,  how  many  dozen 
can  be  bought  for  4'15/  ? 

6.  If  |-  ton  of  coal  lasts  1  week,  how  long  will 
1  ton  last  ? 

7.  If  1  Ib.  of  butter  lasts  a  family  1  week,  what 
part  of  a  week  will  f  of  a  Ib.  last  ? 

Show  pupils  units  that  represent  1,  2,  3,  4. 


1.    If  we  callJ.  1,  what  is   (7?     What  is 
What  is  B  ? 


If  pupils  cannot  give  the  names  (J,  J,  f ,  1),  tell  them. 

2.  Show  me  the  1.     The  f     The  f     The  f 

3.  What  is  the  name  of  the  largest  unit?     Of 


102  PRIMARY  ARITHMETIC. 

the  smallest  ?     Of  the  next  to  the  smallest  ?     Of 
the  next  to  the  largest  ? 

4.  What  are  the  names  of  these  units  ? 

5.  Pat  %  and  |  together.     The  sum  of  £  and  £ 
equals  what  ? 

6.  Put  |  and  f  together.     The  sum  of  £  and  f 
equals  what  ? 

7.  What  part  of  the  unit  1  is  as  large  as  the  J  ? 

8.  What  part  of  the  J  equals  the  |  ? 

9.  Show  the  part  of  the  f  that  equals  the  %  ? 
What  part  of  the  f  equals  the  ^  ? 

10.  |  and  what  equal  f  ? 

11.  1  is  how  much  more  than  ^  ? 

12.  |  and  what  equal  1  ? 

13.  The  unit  1  is  how  many  times  as  large  as 


14.  •  The  unit  1  is  how  many  times  as  large  as 
the  unit  %  ? 

15.  The  unit  ^  is  how  many  times  as  large  as 
the  unit  £  ? 

16.  If  the  ^  weighs  5  oz.?  how  many  5-oz.  do 
each  of  the  other  units  weigh  ? 

Eeview  without  observing  the  units. 

Building.  —  1.    Build  a  unit  equal  to  |-  of  this 
one. 

2.  Build  another  equal  to  J. 

3.  Another  equal  to  ^,     Another  equal  to  |. 
Show  a  different  unit. 


PRIMARY    ARITHMETIC.  103 

4.  Build  a  unit  equal  to  |  of  this  one. 

5.  Build  the  £.     Build  the  f     Build  the  f . 

Cutting.  —  Give  each  pupil  a  rectangle.     Call  it  1. 
Cut  another  1.  .  Cut  £.     Cut  £.     Cut  f . 

Relation  of  gallon  and  quart.  —  Eeview  lesson  on  quart 
and  pint.     Have  pupils  practise  rilling  the  gallon  measure. 


Empty  it.  Fill  it  £  full.  Empty  it.  Fill  it  £  full. 
Empty  it.  Fill  it  f  full.  After  measuring,  have  pupils 
tell  all  they  can  about  the  gallon  and  quart. 

1.  What  is  sold  by  the  gallon  ? 

2.  A  gallon  is  how  many  times  as  much  as  a 
quart  ? 

3.  What  part  of  a  gallon  equals  1  qt.  ? 

4.  If  you  should  make  2  equal  parts  of  a  gallon, 
how  many  quarts  would  there  be  in  each  ?     How 
many  quarts  in  |  gal.  ? 

5.  A  gallon  is  how  much  more  than  a  quart  ? 


104  PRIMARY    ARITHMETIC. 

6.  How  many  quarts  must  be  added  to  half  a 
gallon  to  make  a  gallon  ? 

7.  1  qt.  equals  what  part  of  a  gallon  ? 

8.  If  we  call  a  quart  1,  what  ought  we  to  call 
a  gallon  ?     If  we  call  the  gallon.  1,  what  ought  we 
to  call  the  quart  ? 

9.  A  quart  equals  how  many  pints  ?     A  gallon 
measure  will  hold  how  many  quarts  ?     A  gallon 
measure  will  hold  how  many  2-pts.  ? 

10.  In  3  gals,  there  are  how  many  4-qts.  ? 

Relation  of  gallon,  quart,  and  pint.  —  1.  If  a  pint 
of  water  weighs  1  lb.,  how  much  does  a  gallon 
weigh  ? 

2.  If  a  quart  of  milk  costs  6/?  what  part  of 
6/  will  a  pint  cost  ? 

3.  If  a  quart  of  milk  costs  6/,  how  many  6-/ 
will  a  gallon  cost  ? 

.4.  If  1  gal.  of  milk  costs  24^  ?  what  part  of  24/ 
will  1  qt.  cost? 

5.  If  \  of  a  gal.  of  milk  costs  6^,  how  many  6-/ 
will  a  gallon  cost?     How  many  6-/  will  |-  of  a  gal. 
cost  ?     How  many  6-/  will  f  of  a  gallon  cost  ? 

6.  The  cost  of  3  quart  boxes  of  berries  is  25/. 
The  cost  of  4  boxes  equals  how  many  thirds  of 
25/? 

Relations  of  magnitude. — Place  units  representing  2, 
4,  6,  8  where  they  can  be  handled.  Teach  the  names  2,  4, 
6,  8.  If  the  children  know  the  number  relations  of  2,  4, 
6,  and  8,  use  letters  instead  of  numbers. 


PRIMARY    ARITHMETIC.  105 

1.  What  is  the  name  of  the  smallest  unit  ?     Of 
the  largest  ?     What  is  the  name  of  the  smaller  of 
the  other  two  ?     Of  the  larger  ?     Name  the  units 
in   order,   beginning   with   the    smallest.      Name 
them  in  order,  beginning  with  the  largest. 

2.  Put  two  units  together  and  tell  what  the 
sum  equals. 

Ex. :  4  and  2  equals  6. 

3.  Tell  how  much  less  one  unit  is  than  another. 

4.  The  sum  of  6  and  2  equals  what  ?     The  sum 
of  4  and  4  equals  what  ?     The  sum  of  4  and  2 
equals  what  ? 

5.  4  less  2  equals  what  ?     8  less  4  equals  what? 
6  less  4  equals  what  ?     8  less  2  equals  what  ?     2 
and  what  equal  4  ?     4  and  what  equal  8  ?     4  and 
what  equal  6  ?     2  and  what  equal  8  ? 

6.  How  many  2's  in  each  unit  ? 

7.  Make  sentences  like  this :  2  equals  ^  of  4. 

8.  Make  sentences  like  this  :  8  equals  4  times  2. 

9.  Tell  the  part  of  4,  of  6,  of  8?  that  is  as  large 
as  2.     Tell  the  part  of  6  and  of  8  that  is  as  large 
as  4.     Tell  the  part  of  8  that  is  as  large  as  6. 

10.  4  equals  how  many  times  2  ?      4  equals 
what  part  of  6  ?     Of  8  ? 

11.  6  equals  how  many  times  2  ?     It  equals 
how  many  times  ^  of    4  ?     It  equals  how  many 
times  |  of  8  ?    6  is  3  times  as  large  as  what  unit  ? 

12.  8  equals  how  many  times  2  ?     How  many 
times  ^  of  4  ?     How  many  times  ^  of  6  ? 


106  PRIMARY    ARITHMETIC. 

13.  2  is  how  many  times  as  large  as  1  ?     4  is 
how  many  times  as  large  as  2  ?     8  is  how  many 
times  as  large  as  4  ? 

14.  What  unit  is  2  times  as  large  as  1  ?    As  2  ? 

As  4? 

1.  Find  other  sets  of  solids  that  may  be  called 
2,    4,  6,  8.     Tell  all  the  relations  that  you  can. 

2.  Find  surfaces  that  may  be  called  2,  4,  6,  8. 
Tell  the  relations. 

3.  Find  edges  that  may  be  called  2,  4,  6,  8. 
Tell  the  relations. 

4.  Make  statements  like  this  :  If  we  call  this  a 
2,  we  should  call  this  4,  for  it  is  2  times  as  large 
as  the  2. 

5.  Make  statements  like  this :  If  this  is  8,  then 
this  is  4,  for  it  equals  ^  of  8. 

6.  Call  the  blackboard  8.     Show  the  part  that 
is  as  large  as  4.     As  large  as  2.     As  large  as  6. 

7.  Make  statements  like  this  :  If  we  call  the 
edge  of  this  table  2,  we  must  have  an  edge  2  times 
as  long  if  we  wish  to  call  it  4. 

Cutting.  —  1.  This  is  a  2.  Cut  a  2,  a  4,  a  6, 
and  an  8.  Measure  to  see  if  you 
have  made  each  unit  the  right 
size. 

2.  Try  again.  Cut  a  large 
rectangle  and  call  it  2.  Cut  a  4, 
a  6,  and  an  8.  Measure. 


PRIMARY    ARITHMETIC.  107 

Drawing.  —  1.  Draw  a  rectangle.  Call  it  2. 
Draw  a  4,  a  6,  and  an  8. 

Problems.  —  1.  If  you  can  clean  the  blackboard 
in  8  minutes,  what  part  of  the  board  can  you  clean 
in  4  minutes  ?  In  2  minutes?  In  6  minutes? 

2.  The  money  that  you  pay  for  4  apples  equals 
what  part  of  the  money  that  you  pay  for  6  apples? 
For  8  apples  ? 

3.  If  2  Ibs.  of  candy  cost  $1,  how  much  will 
'8  Ibs.  of  candy  cost? 

4.  How  many  times  as  long  will  it  take  to  walk 
8  miles  as  to  walk  2  miles  ? 

5.  If  it  takes  2  hours  to  walk  8  miles,  how  long 
will  it  take  to  walk  4  miles  ? 

6.  6  yds.  of  ribbon  will  cost  how  many  times  as 
much  as  2  yds.  ?     The  cost  of  4  yds.  equals  what 
part  of  the  cost  of  6  yds.  ? 

7.  1  yd.  of  ribbon  will  cost  how  many  times  as 
much  as  ^  yd.  ?     How  many  times  as  much  as 
iyd.? 

8.  A  gallon  measure  holds  how  many  times  as 
much  as  a  quart  ? 

9.  If  a  quart  of  molasses  costs  a  dime,  how 
many  dimes  will  a  gallon  cost  ? 

10.  6  baskets  of  apples  cost  75/.     What  part 
of   75/  will  2  baskets  cost?     What  part  will  4 
baskets  cost  ? 

11.  8  hours  equals  how  many  thirds  of  6  hours? 
The  distance  you  can  walk  in  8  hours  equals  how 


108  PRIMARY    ARITHMETIC. 

many  thirds  of  the  distance  you  can  walk  in  6 
hours  ? 

Comparing  surfaces Give  each  pupil  a  square  2  in. 

long,  a  square  4  in.  long,  and  a  rectangle  2  in.  by  4  in.,  and 
one  2  in.  by  6  in.,  or  draw  figures  on  the  blackboard  of  the 
same  relative  size. 

1.  Use  the  small  square  as  a  measure  and  tell 
what  you  can   about  the   relations   of   the  rect- 
angles. 

2.  Teach  the  names  A,  B,  C9  and  JD.     If  we  call 
A  2,  what  ought  we  to  call  each  of  the  others  ? 

3.  Call  A  1 ;  what  is  the  name  of  each  of  the 
others  ? 

4.  Call  A  i ;  what  ought  we  to  call  each  of  the 
others? 

5.  If  A  is  ^,  how  many  fourths  in  each  of  the 
others  ? 

6.  If   C  is   1,  A  equals  what  part  of  1  ?     B 
equals  what  part  of  1  ?     D  is  how  many  times  as 
large  as  the  third  of  1  ? 

7.  Call  D  1-,  B  equals  what  part  of  another  1  ? 
A  equals  what  part  ?     C  equals  what  part  ? 

8.  If  C  is  3,  what  is  .£?     What  is  A  ?     What 
isD? 

9.  If  you  can  make  3Ts  of  A,  how  many  Ts 
can  you  make  of  B  ?     Of  C  ?     Of  D  ? 

10.  What  is  the  length  of  A  ?     How  many  2-in. 
in  the  perimeter  of  A  ?     Of  B  ?     Of  D  ? 


PRIMARY    ARITHMETIC.  109 

Ratios  of  length. —  1.  Practise  drawing  a  foot. 
Practise  placing  points  1  ft.  apart.  Try  to  draw 
lines  a  foot  long  with  eyes  closed. 
Measure.  With  your  eyes  closed  try 
to  place  points  1  ft.  apart.  Measure. 

2.  Draw  a  foot.    Draw  a  line  equal 
to  |-  ft.     To  |  ft.     To  |  ft.     Practise 
drawing    and    measuring    groups   of 
these  lines. 

3.  If  we  call  the  shortest  line  1,  what  ought  we 
to  call  each  of  the  other  lines  ? 

4.  If  the  shortest  line  is  ^,  what  is  each  of  the 
other  lines  ? 

Ans. :   1,  f,  2. 

5.  If  the  shortest  line  is  £,  what  is  each  of  the 
other  lines  ? 

6.  Call  the  next  to  the  shortest  line  1 ;  what  is 
the  name  of  each  of  the  other  lines  ? 

7.  If  the  shortest  line  is  £,  find  the  1.     If  the 
longest  line  is  1,  find  |. 

8.  If  the  longest  line  is  ^  what  part  of  ^  is 
each  of  the  other  lines  ? 

9.  Call  the  longest  line  12  ;  what  part  of  12  is 
each  of  the  other  lines  ? 

10.  Call  the  shortest  line  3;   how  many  3's  in 
each  of  the  other  lines  ? 


110 


PRIMARY    ARITHMETIC. 


Have  pupils  assign  different  values  to  the  units  and  tell 
what  the  other  units  are.  Ex. :  Call  A  1 ;  what  is  each  of 
the  others  ? 


Have  pupils   compare  different   units   with   the   other 
units.     Ex.  :  A  equals  2  times  B,  f  of  (7,  |  of  D,  etc. 

Draw  lines  on  the  blackboard  1  ft.,  9  in.,  6  in.,  3  in. 
long.     Teach  the  names  of  the  lines. 

1.  What  is  the  name  of  the  long- 
est line  "I     Of  the  shortest  ?     Of  the 
line  that  is  -J  ft.  long  ?     Of  the  line 
that  is  f  ft.  long  ? 

2.  Name  the  lines  in  order,  be- 
ginning with  the  shortest.     Repeat, 

Name  in  order,  beginning  with  the  longest. 


ARITHMETIC.  Ill 

3.  Make  sentences  like  this :  The  sum  of  6  in. 
and  3  in.  equals  9  in. 

4.  The  3-in.  line  equals  what  part  of  each  of 
the  other  lines  ? 

5.  The  6 -in.  line  equals  how  many  times  the 
3-in.  line  ?     It  equals  what  part  of  each  of  the 
other  lines  ? 

6.  Compare   the    6-in.    line  with  each  of    the 
other  lines  again. 

7.  The  9-in.  line  equals  how  many  times  the 
3-in.  line  ?     It  equals  how  many  halves  of  the  6-in. 
line  ?     It  equals  what  part  of  the  foot  ? 

8.  The  ft.  equals  how  many  times  the  3-in.  line  ? 
It  equals  how  many  times  the  6-in.  line  ?     Show  ^ 
of  the  9-in.  line.     The  ft.  is  as  long  as  how  many 
thirds  of  the  9-in.  line  ?   It  equals  how  many  thirds 
of  the  9-in.  line  ? 

9.  A  foot   is   how   many  times  as  long  as  6 
inches  ?    A  foot  is  how  much  longer  than  6  inches  ? 
6  inches   equal  what  part  of  a  foot  ?     4  inches 
equal  what  part  of  a  foot  ?     A  foot  equals  how 
many  4-inches  ?     What  part  of  a  foot  is  as  long 
as  8  inches  ? 

10.  What  is  J  ft.  ?  What  is  %  ft.  ?  What  is 
f  ft.  ?  What  is  i  of  6  in.  ?  What  is  |  of  9  in.  ? 
What  is  f  of  9  in.  ?  What  is  f  of  a  ft.  ?  Picture 
the  lines  in  your  mind  and  tell  all  you  can  about 
them. 

Review  without  observing  the  lines. 


PRIMARY    ARITHMETIC. 


Problems.  —  1.  If  it  takes  all  your  money  to  pay 
for  a  loaf  of  bread  the  size  of  B,  what  part  of  your 
money  will  it  take  to  pay  for  a  loaf  the  size  of  D  ? 
Of  C  ?  Could  you  pay  for  a  loaf  the  size  of  A  ? 
The  money  you  have  would  pay  for  a  loaf  three 
times  as  large  as  what  part  of  A  ? 


v  2.  |  of  B  is  worth  a  nickel ;  B  is  worth  how 
many  nickels  ?  A  is  worth  how  many  nickels  ? 
For  C  you  must  pay  how  many  times  as  much  as 
for  I  of  C  ?  As  for  \  of  A  ?  The  cost  of  D  equals 
what  part  of  the  cost  of  each  of  the  other  units  ? 

3.  Call  the  blocks  cakes.    If  C  is  enough  for  six 
people,  A  is  enough  for  how  many  people  ?    D  for 
what  part  of  twelve  people  ?     If  D  is  enough  for 
three  people,  B  will  supply  how  many  ? 

4.  One  dollar  will  buy  8  First  Readers.     What 
part  of  one  dollar  will  pay  for  6  First  Readers  ? 
For  4  ?     For  2  ? 

5.  A  gallon  of  oil  will  cost  how  many  times  as 
much  as  \  of  a  gallon  ? 

6.  The  cost  of  2  Ibs.  of  raisins  equals  what  part 
of  the  cost  of  8  Ibs.  ? 


PRIMARY    ARITHMETIC. 


113 


1  doz. 

oooooo 
o  o  o  o  o  o 


1  doz. 
oooo 

0000 

oooo 


1.  How  many  6's  in  a  doz.  ? 

2.  How  many  4's  ?     3's  ?     2's  ? 

Give  each  pupil  a  rectangle  3  in.  by  4  in.,  another  2  in. 
by  4  in.,  and  a  third  1  in.  by  4  in. 


1.  If  M  is  a  doz.,  what  part  of  a  doz.  is  each  of 
the  other  units  ? 

2.  Show  ^  of  a  doz.     f  of  a  doz.     f  of  a  doz. 

3.  A  doz.  is  how  many  times  as  large  as  |  of 
a  doz.  ? 

4.  M  equals  how  many  halves  of  D  ? 

5.  A  doz.  equals  how  many  halves  of  f  of  a  doz.  ? 
Call  C  4  ;  what  is  D  ?     What  is  Jf  ? 

6.  4  equals  what  part  of  8  ?     Of  a  doz.  ? 

7.  8  equals  how  many  times  4  ?    It  equals  what 
part  of  a  doz.  ? 

8.  A  doz.  equals  how  many  times  4  ?     It  equals 
how  many  halves  of  8  ?     Which  is  the  more,  4  or 
JofS? 


114  PRIMARY    ARITHMETIC. 

9.  What  two  equal  units  in  8  ?  What  three 
equal  units  in  a  doz.  ? 

Keview,  using  the  rectangles.     Eeview  without  them. 

Problems.  —  1.  A  doz.  oranges  will  cost  how 
many  times  as  much  as  6  oranges  ?  As  4  ? 

2.  The  cost  of  9  oranges  equals  what  part  of 
the  cost  of  a  doz.  ? 

3.  How  many  3's  in  a  doz.  ?     How  many  4's  ? 
How  many  6's  ? 

4.  If  6  pens  cost  a  dime,  how  many  dimes  will 
a  doz.  pens  cost  ? 

5.  One   half   doz.  pears   will    cost   how   many 
times  as  much  as  ^  of  a  doz.  ? 

6.  A  doz.  eggs   cost  15/.     4  eggs   cost   what 
part  of  15/  ?     8  eggs  cost  what  part  of  15/  ? 

7.  A  doz.  bananas  will  cost  how  many  times  as 
much  as  4  bananas  ? 

8.  The  cost  of  |  of  a  doz.  pencils  equals  what 
part  of  the  cost  of  f  of  a  doz.  ? 

9.  The  cost  of  8  buttons  equals  what  part  of  the 
the  cost  of  a  doz.  buttons  ? 

10.  One  doz.  is  how  many  more  than  8  ?     4  is 
how  many  less  than  one  doz.  ? 

11.  6  and  what  equal  a  doz.  ?    8  and  what  equal 
a  doz.  ?     4  and  what  equal  a  doz.  ?     3  and  what 
equal  a  doz.  ?     9  and  what  equal  a  doz.  ? 

Relation  of  rectangles.  —  Give  each  pupil  a  square  rect- 
angle 3  in.  long,  a  rectangle  3  in.  by  4  in.,  a  rectangle 
2  in.  by  3  in.,  a  rectangle  1  in.  by  3  in. 


PRIMARY    ARITHMETIC. 


115 


1.   What  are  the  names  of  the  rectangles  in  the 
order  of  their  size  ? 


2.  Tell  all  you  can  about  the  rectangles. 

3.  If  H  represents  a  doz.?  what  part  of  a  doz. 
does  each  of  the  others  represent  ? 

4.  The  sum  of  B  and  D  equals  what  unit  ?     It 
equals  what  part  of  a  doz.  ? 

5.  B  equals  what  part  of  each  of  the  other  units  ? 

6.  What  is  the  relation  of   D  to  each  of   the 
other  units  ?     Of  C  ?     Of  HI     Of  the  doz.  ? 

7.  If  B  is  3,  what  is  each  of  the  other  units  ? 

8.  How  many  3's  in  6  ?    In  9?    In  12,  or  a  doz.  ? 

9.  What  is  the  relation  of  3  to  each  of  the  other 
units?     Of  6?     Of  9?     Of  12?     Of  a  doz.  ? 

10.  If  H  is  worth  10/,  what  part  of  10/  is  each 
of  the  others  worth  ? 

11.  If  B  cost  5/?  what  is  the  cost  of  each  of  the 
others  ? 

12.  If    3   cost   5/,  how  many   5/  will   6   cost? 
How  many  5/  will  9  cost  ?     1  doz.  ? 

Eeview,  using  the  rectangles.     Eeview  without  them. 
Drill. 


116  PRIMARY    ARITHMETIC. 

Ratios  of  time.  —  1.  How  long  is  it  from  Christ- 
mas to  the  next  Christmas  ?  From  one  birthday 
to  the  next  ? 

2.  Draw  two  lines,  one  representing  a  yr.,  and 
the  other  J  yr.,  or  6  mos. 

3.  Tell  all  you  can  about  the  yr.  and  6  mos. 


4.  How  many  6-mos.  in  a  yr.  ?     What  part  of  a 
yr.  equals  6  mos.  ?     1  yr.  and  6  mos.  equal  how 
many  6-mos.  ? 

5.  How  many  6-mos.  in  1^  yrs.  ?     6  mos.  equal 
what  part  of  1^  yrs.  ?     1  yr.  equals  what  part  of 
l^r  yrs.  ?     How  many  6-mos.  in  |  of  a  yr.  ? 

Draw  three  lines,  one  representing  a  year,  one  -J-  of  a 
year,  and  the  other  f  of  a  year. 

1.  If  the  shortest  line  represents  4  mos.,  how 
many  4-mos.  does  each  of  the  other  lines  represent  ? 
How    many   months    does 

each  of  the  other  lines  re- 
present ?  How  many  thirds 
— —  of  a  yr.  does  each  of  the 

lines  represent  ? 

2.  Compare  4  mos.  with  8  mos. ;  with  a  yr. 

3.  Compare  8  mos.  with  4  mos. ;  with  a  yr. 

4.  Compare  1  yr.  with  4  mos. ;  with  8  mos. 


PRIMARY    ARITHMETIC.  117 

Problems.  —  1.  The  money  that  Harry  can  earn 
in  6  mos.  equals  what  part  of  the  money  that  he 
can  earn  in  a  yr.  ?  In  8  mos.  ? 

2.  The  number  of  months  in  1  yr.  equals  how 
many  times  the  number  in  J  yr.  ?     In  ^  yr.  ? 

3.  The  number  of  days  in  3  mos.  equals  what 
part  of  the  number  in  4  mos.  ?     In  6  mos.  ?     In 
9  mos.?     In  1  yr.  ? 

4.  One  yr.  is  how  much  longer  than  8  mos.  ? 

5.  The  time  from  New  Year  to  New  Year  equals 
how  many  halves  of  8  mos.  ? 

Relation  of  dime  and  nickel.  —  1.  If  A  represents 
a  dime,  what  is  the  name  of  the  piece  of 
money  represented  by  B  ? 

2.  How  many  nickels  equal  1  dime  ? 

3.  A  dime  and  a  nickel  equal  how  many 
nickels  ? 

4.  A  nickel  equals  what  part  of  a  dime  ? 

5.  The  candy  you  can  buy  for  a  nickel 
equals  what  part  of  the  candy  you  can  buy  for  a 
dime? 

6.  A  nickel  equals  how  many  cents  ? 

7.  A  nickel  and  how  many  cents  equal  a  dime  ? 

8.  A  dime  equals  how  many  5/  ? 

9.  5/  equals  what  part  of  a  dime  ? 

10.  A  dime  and  5/  equals  how  many  5/  ? 

11.  1^-  dimes  equal  how  many  5/  ? 

12.  5/  equals  what  part  of  1^  dimes  ? 

13.  A  dime  equals  what  part  of  1 J  dimes  ? 


118  PRIMARY    ARITHMETIC. 

Relative  values.  -  -  1.  If  the  shortest  line 
represents  2/,  what  do  each  of  the  other  lines 
represent  ? 

2.  Point  to   the  different  lines   and  tell  what 
each  represents. 

3.  2/  equals  what  part  of  4/  ?   Of  6/  ?   Of  8/  ? 
Of  a  dime  ? 

4.  Compare  4/  with  each  of  the  other  units. 
Compare  6/  with  each.     Compare  8/  with  each. 
Compare  a  dime  with  each. 

Problems.  —  1.   2/    will    buy    an     apple.      4/ 

will   buy   how  many   apples? 

. How  many  will  6/  buy  ?     A 

dime  ? 

2.   A  boy  sells  papers  for  2/ 
each.     How  many  does  he  sell 
2ft  to  receive  a  dime  ? 

3.  2/   is   ^   of    Nellie's  money.      How   much 
money  has  she  ? 

4.  John  has  10/  and  loses  ^  of  it ;  how  much 
does  he  lose  ?     How  many  2/  has  he  left  ? 

5.  4  peaches  equal  what  part  of   6  peaches  ? 
Of  8?     Of  10? 

6.  f  of  a  Ib.  of  cheese  cost  7/.     How  many  7/ 
will  |  of  a  Ib.  cost  ? 

7.  If  f  of  a  doz.  pencils  cost  6/?  what  is  the 
cost  of  a  doz.  ? 


PRIMARY    ARITHMETIC. 


119 


Ratios  of  solids.  —  Place  solids  which  represent  1,  2, 
3,  4,  5  where  they  can  be  handled. 


1.  Learn  the  names  A,  B,  C,  D,  and  E. 

2.  Tell  all  you  can  about  these  units.     Tell  all 
you  can  about  these  units  without  looking  at  them. 

3.  Unite  different  units  and  tell  what  they  equal. 
Ex. :  The  sum  of  A  and  B  equals  C. 

4.  Make  statements  like  this :  E  less  A  equals  D. 

5.  B  and  what  equals  C  ?     A  and  what  equals 
C  ?     C  and  what  equals  D  ?    B  and  what  equals  D  ? 

6.  Look  at  B.   How  many  As  equal  B  ?    What 
part  of  C  equals  B  ?     What  part  of  D  equals  B  ? 
What  part  of  E  equals  B  ? 

7.  D  equals  two  times  what  unit  ?     It  is  two* 
times  as  large  as  what  part  of  C  ?    It  is  two  times 
as  large  as  what  part  of  E  ? 

8.  What  part  of  C  is  two  times  as  large  as  A  ? 
What  part  of  D  is  two  times  as  large  as  A  ? 


120  PRIMARY    ARITHMETIC. 

9.    B  equals   how   many  times  At     It   equals 
what  part  of  each  of  the  other  units  ? 

10.  C  equals  how  many  times  ^  of  Dt     How 
many  times  ^  of  B  ?      It  equals  what  part  of  each 
of  the  other  units  ? 

11.  D  equals  how  many  times  At     D  equals 
how  many  times  Bt     D  is  how  many  times  as 
large  as  f  of  C  t     D  equals  two  times  what  part 
of  Et 

1.  E  equals  D  and  what  part  of  another  Dt 
E  equals  C  and  what  part  of  another  C  ?   E  equals 
how  many  B's  ?    Am. :  E  equals  2|-  j?'s.    E  equals 
how  many  ^L's  ? 

2.  Call  E  1 ;    each  of  the  other  units  is  what 
part  of  another  1  ? 

3.  Call  D  1 ;  what  is  each  of  the  other  units  ? 

4.  Call  C  1 ;  what  is  each  of  the  other  units  ? 

5.  Call  A  1 ;    each  of   the  other  units  equals 
how  many  times  1  ? 

6.  Call  A  2  ;  what  is  each  of  the  other  units  ? 

7.  Call  A  I  ;   what  is  each  of  the  other  units  ? 
Ans. :  B  is  £,  C  is  f ,  D  is  1  and  E  is  1  \. 

8.  Call  A  | ;  what  is  each  of  the  other  units  ? 

9.  Call  A  \  ;  what  is  each  of  the  other  units  ? 
*  10.   Call  A  3  ;  what  is  each  of  the  other  units  ? 


PRIMAKY    ARITHMETIC. 


121 


Drawing  and  cutting.  —  1.  Draw  a  line.  Divide 
it  so  that  one  part  will  represent  the  unit  2,  and 
the  other  the  unit  3.  Measure. 

2.  Draw  a  rectangle.      Divide  it  so  that  one 
part  will  represent  2,  and  the  other  3.     Measure. 
Practise. 

3.  Cut  rectangles.      Divide  them  so  that  they 
will  represent  the  unit  2  and  3.    Measure.    Practise. 
Practise  drawing,  dividing,  and  measuring. 


Relative  areas.  —  1.  Cut  the  units  2,  4?  3,  5,  1,  2. 

2.  Measure1  each  by  2,  and  tell  how  many  2's 
in  each. 

Ex. :  In  5  there  are  2f  2's.    In  1  there  is  £  of  2. 

3.  Tell  how  much  more  one  unit  is  than  another. 
Ex. :  4  is  three  more  than  1.  4  is  two  more  than  2. 

4.  Tell  how  much  less  one  unit  is  than  another. 

5.  Unite  units  and  tell  what  the  sum  equals. 
Ex. :  The  sum  of  2  and  1  equals  3. 

1  First,  make   estimates  with   the  eye.     Afterward  test  judg- 
ments by  using  a  measure. 


122  PRIMARY    ARITHMETIC. 

6.  What  two  units  equal  4  ?     What  other  units 
equal  4  ? 

7.  What  two  units  equal  5  ?     What  other  units 
equal  5  ?     What  three  units  equal  5  ? 

8.  If  the  unit  A  is  1,  each  of  the  other  units 
equals  how  many  halves  ? 

9.  Compare  each  unit  with  the  unit  2. 

10.  If  the  unit  2  is  worth  a  dime,  what  is  each 
of  the  other  units  worth  ? 

11.  Draw  units,  making  each  two  times  as  large 
as  the  2,  4,  3,  etc.     Measure  to  see  if  you  have 
made  the  units  two  times  as  large.     Write  the 
names  2,  4?  etc. 

12.  Tell  all  you  can  about  the  units  you  have 
drawn. 

Eeview. 

"The  starting  point  is,  constantly  and  necessarily,  the 
knowledge  of  the  precise  relations,  i.e.  of  the  equations, 
between  the  different  magnitudes  which  are  simultaneously 
considered."  -  —  Comte. 

1.  The  unit   3    is   how  much  more   than   the 
unit  1  ?    1  is  how  much  less  than  3  ?    3  apples  are 
how  many  more  than  1  apple  ? 

2.  4  is  how  much  greater  than  2  ?      2  and  2 
equal  what  ?     4  is  how  many  times  as  large  as  2  ? 
2  equals  what  part  of  4  ? 

3.  5  is  how  much  greater  than  1  ?     Than  3  ? 
Than  2  ?     Than  4  ?     What  must  be  added  to  3  to 
make  5?     To  1  to  make  5?     To  2  to  make  5? 


PRIMARY    ARITHMETIC. 


123 


5  pens  are  how  many  more  than  3  pens  ?     Than 
2  pens  ? 

4.  The  sum  of   3  and  2  equals  what?      Of  1 
and  4  ?     Of  2  and  2  and  1  ?     Of  1  and  2  and  2  ? 
Of  3  and  2  ? 

5.  Henry  paid  3/  for  candy  and  2/  for  nuts ; 
how  much  did  he  pay  for  both  ? 

6.  Nellie  spent  5/  for  pears  and  2/  for  pins; 
how  much  more  did  she  pay  for  the  pears  than  for 
the  pins  ? 

Separating  and  combining.  —  1.  How  many  1's 
do  you  see  in  this  diagram  ?  How  many  2's  ? 
How  many  3's  ?  How  many  4's  ?  5's  ? 


2.  If  d  is  2,  what  is  the  name  of  the  units 
under  each  letter  ? 

3.  Unite  the  two  units  under  each  letter  and 
think  the  unit  to  which  the  sum  is  equal. 

Ex. :  Look  at  the  units  under  e  and  think  Jf,. 

4.  Look  at  diagram  and  name  sums. 


124  PRIMARY    ARITHMETIC. 

Ex. :  Look  at  the  two  units  under  e  and  say  £. 

5.  Draw   units   on   the   blackboard    and    have 
pupils  practise  thinking  sums. 

6.  After   observing    the    units    carefully,    turn 
away  from  them  and  pronounce  the  sums  under 
each  letter. 

The  expression  for  quantitative  ideas  should  be  acquired 
as  the  everyday  vocabulary  has  been,  —  by  repeatedly 
bringing  into  consciousness  the  relations  which  the  terms 
express. 

As  the  pupil  advances,  sight  forms  should  suggest  ideas, 
just  as  spoken  words  do.  But  as  reading  should  be  ap- 
proached through  sense  training,  an  interest  in  things,  and 
the  power  to  talk  freely,  so  should  the  use  of  written 
forms  in  mathematics.  At  the  proper  time  the  teacher 
should  find  occasion  to  present  the  written  expression 
freely  and  in  such  manner  that  the  primary  attention  is 
still  held  to  the  relations  discerned.  Gradually  the  use  of 
language  in  mathematics  should  become  as  automatic  as 
the  use  of  language  in  other  subjects. 

The  principles  which  govern  practice  in  aiding  a  child' 
to  think  in  symbols  apply  in  mathematics  as  elsewhere. 
For  example,  when  we  wish  to  acquaint  the  child  with  the 
written  symbols  for  his  thought  of  the  color  of  a  black 
dog,  we  write,  "  The  dog  is  black."  So,  when  a  pupil  tells 

3 
you  that  3  and  2  equal  5,  write  2  so  that  his  eye  may  take 

5 

in  the  expression  as  a  whole.  We  should  represent  the 
complete,  not  the  partial  thought  of  the  pupil,  —  the 
equation,  not  a  part  of  it.  Fix  the  thought  so  firmly 
that  finally  one  side  of  the  equation  will  suggest  the 
other. 


PRIMARY    ARITHMETIC. 


125 


Ask  the  following  questions,  and  write  answers  on  the 
blackboard  :  — 


3  and  1  equal  what  ?  a         b 

1  and  2  equal  what  ?  31 

2  and  2  equal  what  ?  _?         1 
2  and  3  equal  what  ?  52 

3 


Answers, 
c 
2 
2 
4 


d 

2 
1 
3 


e 
1 
3 
4 


1 

Observe  1 : 
2 

2 


image ;  write ; 
practise. 


Observe  2;  image;  write; 
4         practise. 


Observe  1  2 ;  image; 

5     write. 

3     T 
Observe  2     1;  image 

5     2     write. 

312 
Observe  212;  image ;  write ;  practise. 

524 

Continue  adding  one  combination  at  a  time,  until  the 
pupils  can  image  and  write  the  five  readily. 

Tell  the  combination  under  each  letter,  thus :  2 
is  under  a.  5 

What  combination  is  under  c  ?  Under  6?  Under  &? 

Show  the  combination  at  the  left.  The  second 
from  the  right.  Image  and  think  each  combina- 
tion with  its  sum,  beginning  at  the  top. 

Image  each  combination  and  pronounce  the  sum. 

1  "  The  habit  of  hasty  and  inexact  observation  is  the  founda- 
tion of  the  habit  of  remembering  wrongly."  —  Dr.  Maudsley. 

"  A  few  such  items  must  be  memorized  and  reviewed  daily, 
adding  a  small  increment  to  the  list  as  soon  as  it  has  become  per- 
fectly mastered."  —  W.  T.  Harris, 


126  PRIMARY    ARITHMETIC. 

Continue  to  work  with  these  five  combinations  until 
they  are  indelibly  fixed. 

31222 
Write  on  blackboard :  22213 

Think  the  sum  of  each.     Pronounce  the  sum. 
Do  not  say  3  and   1  are  4,  nor  3,  1,  4 ;   but 

observe  g  and  say  4. 

Name  sums  from  right  to  left,  without  observ- 
ing the  board.  From  left  to  right.  What  is  the 
second  sum  from  the  right  ?  The  third  from  the 
left?  etc. 

Make  columns  of  the  combinations,  omitting  sums,  thus  : 

abed  Have  pupils  look  at  each 

column  carefully  and  image 

3211  the  sum  of  each  combination 

2213  of  two  figures.  Picture,  slow- 

ly at  first,  the  combinations 

1321  under  a:  5,  2,  4,  3,  4,  then 

1221  more  quickly,  but  not  so 

quickly  as  to  destroy  the 

2213          visual  image. 

2122  It  will  be  easy  to  secure 

rapidity  after  the  habit  of 

2132          imaging  has  been  established. 

1312  Image,1  beginning  at  the 

bottom. 

1131  Image  from  right,    thus  : 

3         1         2         2          4,  2,  4,  5. 

1  "  There  can  be  no  doubt  as  to  the  utility  of  the  visualizing 
faculty  where  it  is  duly  subordinated  to  the  higher  intellectual 


PRIMARY   ARITHMETIC. 


127 


1.  Measure  each  unit  by  2.1     By  3.     By  4. 

2.  If  c  is  2,  what  is  the  name  of  each  of  the 
other  units  ? 

3.  On  each  unit  that  you  draw  or  cut,  write  the 
name. 

4.  Tell  all  you  can  about  these  units. 

f 


5.  What  two  units  are  as  large  as  6  ? 

6.  Into  what  two  equal  units  can  you  separate  4? 
7*  What  two  equal  units  in  6  ?     What  three 

equal  units  in  6  ?  What  two  unequal  units  do  you 
see  in  6  ? 

operations.  A  visual  image  is  the  most  perfect  form  of  mental 
representation  wherever  the  shape,  position,  and  relations  of 
objects  in  space  are  concerned."  —  Francis  Galton. 

"Addition,  as  De  Morgan  somewhere  insisted,  is  far  more 
swiftly  done  by  the  eye  alone  ;  the  tendency  to  use  mental  words 
should  be  withstood."  —  Francis  Galton. 

1  Do  not  permit  counting.  Wait  until  the  pupil  observes  and 
becomes  conscious  of  the  relative  size  of  the  units. 


128  PRIMARY    ARITHMETIC. 

8.  The  unit   7  is  how   much  larger  than  the 
unit  4  ?      Than   the   unit    3  ?      4   is   how   much 
less  than   7  ?     What  must  be  taken  out  of  7  to 
leave  4  ? 

9.  4  and  what  equal  6  ?      2  and  what  equal  6  ? 

10.  4  and  what  equal  7  ?       3  and  what  equal  7  ? 

11.  6  and  what  equal  7  ?      2  and  what  equal  7  ? 

12.  5  and  what  equal  7  ? 

13.  7  cherries  are  how  many  more  than  3  cher- 
ries ?     Than  5  ?     Than  2  ?     Than  1  ? 

14.  The  sum  of  3  and  3  equals  what  ?     6  equals 
how  many  times  3  ?      3  equals  what  part  of  6  ? 
6  is  how  much  greater  than  3  ?     3  is  how  much 
less  than  6  ? 

15.  Cora  paid  5/  for  paper  and  2/  for  a  pencil. 
How  much  did  she  pay  for  both?      How  much 
more  did  she  pay  for  the  paper  than  for  the  pencil  ? 
How  much    more    for  both  than  for  the  paper  ? 
The  cost  of  the  pencil  equals  what  part  of  the  cost 
of  the  paper  ? 

16.  If  C  is  a  rug  containing  2  square  feet,  how 
many  square  feet  in  each  of  the  other  rugs  ? 

17.  Call  C  1.     What  is  the  name  of  each  of  the 
other  units  ? 

18.  If   the  width  of   E  is   1   foot,  what  is  its 
perimeter  ?     How  many  more  feet  in  the  perimeter 
of  E  than  in  the  perimeter  of  C  ? 

19.  Call  C  J.     What  is  the  name  of  each  of  the 
other  units  ? 


PRIMARY    ARITHMETIC. 

Draw  units  on  the  blackboard. 


129 


1.  How  many  1's  do  you  see  in  this  diagram? 
How  many  2's  ?    How  many  3's  ?    How  many  4's  ? 
How  many  5's  ? 

2.  Practise  looking  at  the  diagram  and  thinking 
the  sums. 

3.  Look  and  name  the  sums. 

4.  Think  and  name   sums  without  looking  at 
the  blackboard. 

Ask  the  following  questions,  and  write  answers  on  the 
blackboard  :  — 


3  and  3  equal  what  ? 

2  and  5  equal  what  ? 

4  and  1  equal  what  ? 
4  and  2  equal  what  ? 

3  and  4  equal  what  ? 
(See  method  of  study  on 

pages  125,  126.) 


4 
3 

7 


Answers. 
215 
442 
657 


3 
3 
6 


130 


PRIMARY   ARITHMETIC. 


Use  any  set  of  solids  having  the  relation  of  1,  2,  3,  4,  5. 

1.  If  2  is  the  name  of  the  smallest  unit,  what 
is  the  name  of  each  of  the  others  ? 

2.  Give  the  names  beginning  with  the  smallest 
unit.     Give  the  names  beginning  with  the  largest 
unit. 

3.  Tell  all  you  can  about  these  units. 


10 


4.  Unite  different  units,  and  tell  what  the  sum 
equals. 

5.  Make  sentences  like  this :  8  less  6  equals  2. 

6.  4  and  what  equals  6  ?    2  and  what  equals  6  ? 
4  and  what  equals  8  ?     6  and  what  equals  10  ? 

7.  Tell  what  two  equal  units  are  found  in  each 
unit.     Ex. :  In  the  unit  6  there  are  2'3's. 

8.  How  many  2's  in  each  unit?      Each  unit 
equals  how  many  2's  ? 

9.  What  is  the  relation  of   2  to  each  of   the 
other  units  ? 

Use  another  set  of  solids  having  the  same  relations. 
Name  them  3,  6,  9,  12,  15.  Work  with  these  units  as  you 
did  with  the  2,  4,  6,  8,  10. 


PRIMARY    ARITHMETIC.  131 

Show  a  solid.  Give  it  a  name,  and  ask  pupils  to  find  a 
related  solid. 

Ex.  :  This  is  9;  find  3.     This  is  10  ;  find  2. 

Drawing.  —  Draw  rectangles  having  the  relations  of  4, 
8,  12,  16,  20  on  blackboard.  Work  with  these  units  as  you 
did  with  2,  4,  6,  8,  and  10. 

Draw  a  rectangle.  Give  it  a  name.  Pupils  draw  re- 
lated rectangles.  Ex.  :  This  is  a  12  ;  draw  a  4. 

Draw   rectangles  either   larger  or  smaller   than   these 
abov^e,  but  having  the  same  relations.     Teach  these 
relations  through  the  language  5,  10,  15,  etc. 

Cutting.  —  This  is  a  1.     Cut  a  1,  2,  3,  4,  5. 
This  is  a  2.     Cut  a  2,  4,  6,  8,  10. 
This  is  a  3.     Cut  a  3,  6,  9,  12,  15. 
Give   other   exercises   in   cutting   and   drawing, 
which  will  fix  the  relative  sizes  of  these  units. 

Separating  and  combining.  —  1.  Measure  each 
unit  by  2. 

Ex. :  There  are  If  2's  in  0. 

2.  How  many  of   the  units  contain  an  exact 
number  of  the  2's  ? 

3.  Measure  each  unit  by  3. 

4.  If  A  is  2,  what  is  the  name  of  each  of  the 
other  units  ? 

5.  Tell  all  you  can  about  the  relations  of  these 
units.     Measure  each  by  2. 

Ex.  :  There  are  If  2's  in  3. 

6.  What  units  united  will  make  8  ?     What  two 
equal  units  in  8  ?     What  four  equal  units  in  8  ? 


132 


PRIMARY    ARITHMETIC. 


What  two  unequal  units  in  8  ?     What  other  un- 
equal units  in  8  ? 

1.  The  unit   8  is   how  much  larger  than  the 
unit  6  ?     Than  the  unit  5  ?     Than  3  ?     Than  4  ? 

2.  How  many  4's  in  8  ?     8  is  how  much  more 
than  4  ?     8  equals  how  many  times  4  ?     4  equals 


what  part  of  8  ?     How  many  2's  in  8  ?      In  6  ? 
6  equals  what  part  of  8  ? 

3.  Show  me  the.  unit  equal   to  -f  of  8.     Show 
me  the  unit  equal  to  |  of  6. 

4.  What  is  the  sum  of  4  and  4  ?     Of  6  and  2  ? 
Of  4  and  3  ?     Of  5  and  3  ?     Of  2  and  6  ?     Of  3 
and  5? 

5.  4  and  what  equal  8  ?     6  and  what  equal  8  ? 
2  and  what  equal  5  ?    2  and  what  equal  8  ?    5  and 
what  equal  8  ?    3  and  what  equal  6  ?    3  and  what 
equal  5  ?     3  and  what  equal  8  ? 


PRIMARY   ARITHMETIC. 


133 


6.  A  boy  had  8  marbles  and  lost  ^  of  them. 
How  many  had  he  left  ? 

7.  A  little  girl  had  8  dolls.      She  gave  £  of 
them  to  some  poor  children.     How  many  did  she 
give  away? 

Draw  the  units  on  the  blackboard. 


Have  pupils  practise  thinking  units  and  suras  under 
each  letter. 

Have  pupils  think  and  name  sums  without  looking  at 
diagram. 

Ask  the  following  questions  and  write  answers  on  the 
blackboard  :  — 

4  and  3  equal  what  ?  Answers. 

5  and  2  equal  what  ?  432         2         3 

6  and  2  equal  what  ?  4        j>         6         5         4 
5  and  3  equal  what  ?  88877 
4  and  4  equal  what  ? 


134 


PRIMARY   ARITHMETIC. 


4 

3 

2 

3 

4 

5 

5 

5 

3 

2 

3 

2 

5 

6 

4 

6 

4 

3 

2 

2 

4 

4 

5 

6 

3 

4 

2 

2 

4 

4 

5 

6 

(See  method  of  study, 
pages  125,  126.) 


Draw  units  on  the  blackboard.1 

1.  Draw  these  units.     Write  the  names  :  thus, 
1,  2  ;   2,  4  ;    3,  6  ;   4,  8  ;    5,  10. 

2.  Tell  all  you  can  about  these  units. 


3.  Make  sentences  like  this :    In   6  there  are 
2'3's. 

4.  The  sum  of  4  and  4  equals  what?     Make 
sentences  like  this :  The  sum  of  4  and  4  equals  8. 

5.  One  half  of  6  equals  what  ?    Make  sentences 
like  this  :  3  =  f .     (Read  :  3  equals  £  of  6.) 

6.  The  sum  of  10  and  5  equals  how  many  5's? 

7.  If  a  10  and  a  5  are  put  together,  the  sum 
equals  how  many  5's  ?     Make  sentences  like  this  : 
The  sum  of  10  and  5  equals  3'5's. 

1  In  all  similar  exercises  the  teacher  should  draw  the  units  on 
the  board,  making  them  of  such  size  that  the  eyes  of  the  pupils 
will  not  be  unduly  taxed  in  observing  them. 


PRIMARY    ARITHMETIC.  135 

8.  Compare  each  of  the  upper  units  with  the 
one  below  it. 

Ex.:  4  =  2  times  2. 

9.  Compare  each  of  the  lower  units  with  the 
one  above  it. 

Tell  everything  you  can  about  the  units  without 
observing  them. 

Practise  thinking  the  sums  of  the  two  units. 

Ex. :  Look  at  1  and  2  and  think  3 ;  at  4  and  2 
and  think  6. 

Think  sums  without  observing  diagram. 

Name  sums  without  observing  diagram. 

Problems.  —  1.  If  3  peaches  cost  a  nickel,  how 
many  nickels  will  6  peaches  cost  ? 

2.  If  6  peaches  cost  a  dime,  what  part  of  a  dime 
will  3  peaches  cost  ? 

3.  4  books  cost  a  dollar;  what  is  the  cost  of 
8  books  ? 

4.  10  Ibs.  of   coffee  cost   how  many  times  as 
much  as  5  Ibs.  ? 

5.  The  cost  of  5  Ibs.  of  coffee  equals  what  part 
of  the  cost  of  10  Ibs.  ? 

6.  The  weight  of  3  Ibs.  of  sugar  equals  what 
part  of  the  weight  of  6  Ibs.  ? 

7.  The  cost  of  a  pt.  of  milk  equals  what  part  of 
the  cost  of  a  qt.  ? 

8.  If  2  apples  cost  4/,  what  part  of  4/  will 
1  apple  cost  ? 


136 


PRIMARY    ARITHMETIC. 


9.  If  2  baskets  of  apples  cost  75/?  what  part  of 
the  75/  will  1  basket  cost  ? 

10.  If  John  can  walk  to  school  in  ^  hour,  how 
long  will  it  take  him  to  walk  to  school  and  home 
again  ? 

Separating  and  combining.  —  In  this  work,  aim  to  secure 
the  association  of  the  three  figures  in  the  mental  picture. 
After  imaging,  test  the  mental  picture  by  having  the 
pupils  supply  from  memory  the  figures  denoting  the  sums. 
Do  this  in  all  similar  exercises. 

2         4         35 
2         4         3       _5 

4         8         6       10 

2  4 

Observe  2  ;  image;  write.  Observe  4  ;  image;  write; 
4  8        practise. 

24  3 

Observe  2     4  ;    image;      Observe  3  ;  image;  write. 
4     8     write.  6 

243 

Observe  2     4     3 ;  image  ;  write  ;  practise. 
486 


Observe  5  ;  image  ;  write  ;  practise. 

10- 


35 


24 

Observe  24     3  _5;  image;  write. 
4     8     6  10 


PRIMARY    ARITHMETIC.  137 

Practise  until  pupils  can  write  the  four  combinations 
easily  and  quickly  from  memory. 

Have  pupils  practise  thinking  the  sums. 

1.  Image  and  pronounce  the  sums  4,  8,  etc. 

2.  What  two  equal  units  in  4  ?     In  8  ?     In  6  ? 
In  10? 

3.  What  is  f  ?     (Read  :  What  is  £  of  4  ?) 

4.  What  is  f  ?     What  is  -\f>-  ?     What  is  f  ? 

2.     4,     3,     5. 

5.  Image  two  of  each  of  the  above  figures,  with 

5 
the  sum.     Ex. :  Image_5  .     Practise. 

10 

Ask  pupils  the  following  questions,  and  write  their 
answers  on  the  blackboard.  After  having  written  answers 
for  several  days  on  the  blackboard,  without  calling  direct 
attention  to  them,  see  if  some  of  the  brighter  pupils  can- 
not read  the  answers.  The  child  should  learn  the  expres- 
sion without  separating  it  from  the  thought. 

Questions.  Answers . 

2  ==  what  part  of    4  ?  2  =  f 

3  =  what  part  of    6  ?  3  =  f . 

4  =  what  part  of    8  ?  4  =  f . 
8  =  what  part  of  10  ?  5  —  ~1/-. 

2  =  how  many  times  1  ?  2  =  2  times  1 . 

4  =  how  many  times  2  ?  4  =  2  times  2. 

6  =  how  many  times  3  ?  6  =  2  times  3. 

8  =  how  many  times  4  ?  8  =  2  times  4. 

10  =  how  many  times  5  ?  10  =  2  times  5. 


138  PRIMARY    ARITHMETIC. 

Draw  on  the  blackboard. 


1.  Tell  all  you  can  about  these  units. 

2.  Measure  each  by  3.     By  2. 

3.  How  many   of    the   units   can    be    exactly 
measured  by  3  ? 

4.  The  unit  A  is  equal  to  what  part  of   (7? 
Of  5?    Of^? 

5.  If  A  is  3,  what  is  the  name  of  each  of  the 
other  units  ? 

6.  Into  what  equal  units  can  you  separate  9  ? 

7.  Into  what  two  equal  units  can  you  separate  6  ? 

8.  What  three  equal  units  in  6  ? 

9.  What  two  unequal  units  in  9  ?     What  other 
unequal  units  in  9  ? 


PRIMARY    ARITHMETIC. 


139 


1.  9  is  how  much  greater  than  6?     Than  7? 
Than  5  ?     Than  4  ? 

2.  5  is  how  much  less  than  9  ?     3  is  how  much 
less  than  9  ?     4  is  how  much  less  than  9  ? 

3.  3  and  what  equal  9  ?     3  and  what  equal  7  ? 
7  and  what  equal  9  ?    5  and  what  equal  7  ?    2  and 
what  equal  7  ?    2  and  what  equal  5  ?     6  and  what 
equal  9  ?     6  and  what  equal  8  ? 

4.  How  many  3's  in  9  ?     3  equals  what  part 
of  9  ?     6  equals  what  part  of  9  ?     9  equals  how 
many  times  3  ?     9  is  how  much  more  than  3  ? 

5.  9/  are  how  many  more  than  5/  ?     A  nickel 
and  how  many  cents  equal  9/  ? 

6.  A  lady  can  dress  3  dolls  in  a  day.      How 
many  can  she  dress  in  3  days  ? 

7.  A  house  has  5  rooms  on  the  first  floor  and 
4  on  the  second.     How  many  rooms  on  the  two 
floors  ? 

Ratios.  —  Draw  the  units  on  the  blackboard. 


12 

U 

16 

20 

18 

1.  Draw  the  units  and  write  the  names. 

2.  Tell  all  you  can  about  these  units. 


140  PRIMARY    ARITHMETIC. 

3.  Compare  each  of  the  upper  units  with  the 
one  below  it. 

4.  Compare  each  of  the  lower  units  with  the 
one  above  it. 

5.  Make  sentences   like  this :    The  sum  of   12 
and  6  equals  3'6's. 

1.  The  number  of  eggs  in  1  doz.  equals  how 
many  times  the  number  in  |-  doz.  ? 

2.  The  number  of  pts.  in   12  qts.  equals  how 
many  times  the  number  in  6  qts.  ? 

3.  The  number  of  pts.  in  6  qts.   equals  what 
part  of  the  number  in  12  qts.  ? 

4.  The  number  of  days  in  7  wks.  equals  what 
part  of  the  number  in  14  wks.  ? 

5.  The  number  of  cents  in   16  nickels  equals 
how  many  times  the  number  in  8  nickels  ? 

6.  The  cost  of  8  bu.  of  potatoes  equals  what 
part  of  the  cost  of  16  bu.  ? 

7.  Eggs  are  10/  a  doz,     How  many  doz.  can 
be  bought  for  20/  ? 

8.  9  ft.  equal  3  yds.     18  ft.  equal  how  many 
3-yds.  ? 

See  work  on  previous  similar  table,  pages  136,  137. 

6       8       7       10       9 

_6  _8     J_       10    _9 

12  16     14       20     18 

1,    What   two  equal    units    in    12  ?       In   20  ? 

In  14  ?     In  16  ?  In  18  ? 


(UNIVERSITY 


PRIMARY    ARITlm^^^UFOR^^^     141 

2.    What  is  -\2-  ?   ( Read  :    What  is   £  of    12  ?) 

6       8       7       10       9 
Image  two  of  each  of  the  above  figures,  with 

9 

the  sum.     Ex. :  Image_9  ;  practise. 

18 

Separating  and  combining.  —  Draw  these  units  on  the 
blackboard. 

a  b  d 


1.  Point  to  each  and  tell  its  name. 

2 .  What  is  the  sum  of  the  units  under  each  letter  ? 

3.  Observe  A  and  think  9. 

4.  Practise  imaging  A9  B,  etc.,  and  think  sum. 

5.  Draw  units  on  blackboard  and  practise  think- 
ing sums. 


142 


PRIMARY   ARITHMETIC. 


Ask  the  following  questions,  and  write  answers  on  the 
blackboard.     See  method  of  study,  pages  125,  126. 


5  and  3 

equal  what? 

6  and  3 

equal  what  ?             4 

2133 

7  and  1 

equal  what  ?             5 

7765 

7  and  2 

equal  what  ?             9 

9898 

5  and  4 

equal  what  ? 

5 

1 

3 

1        7        5 

2         3 

4 

6 

6 

714 

7        6 

3 

3 

1 

522 

4         6 

6 

6 

7 

477 

5        1 

Ratios.  —  Draw  the  units  on  the  blackboard. 

3 

6 

9 

12 

15 

2 

4 

6 

8 

10 

m 

2 

3 

4 

5 

1.  Draw  the  units  and  write  their  names. 

2.  Tell  all  you  can  about  these  units. 

3.  In  each  set  of  three  units,  compare  each  with 
the  other  two.     Ex. :  2  =  |,  |. 

4  =  2  times  2,  *p. 
6  =  3  times  2,  *p. 


PRIMARY    ARITHMETIC.  143 

4.    Tell  everything  you-  can  about  these  units 
without  observing  them. 

1.  At  I/  each,  how  many  postal  cards  can  you 
buy  for  3/  ? 

2.  At  2/  each,  how  much  will  3  postage -stamps 
cost? 

3.  If  Mary  buys  3  rolls  at  2/  each,  how  much 
must  she  pay  ? 

4.  If  each  edge  of  a  triangle  is  2  ft.  long,  how 
many  ft.  in  the  perimeter  of  the  triangle  ? 

5.  If  a  yd.  of  ribbon  costs  3'10/,  how  many  10/ 
will  f  of  a  yd.  cost  ? 

6.  If  a  lady  pays  5/  for  a  ft.  of  picture  framing, 
how  much  ought  she  to  pay  for  a  yd.  ? 

7.  6  yds.  of  cloth  will  make  how  many  times 
as  many  doll's  dresses  as  2  yds.  ? 

8.  The  cost  of  the  cloth  to  make  4  dresses  equals 
what  part  of  the  cost  to  make  6  dresses  ? 

9.  What  is  the  relation  of  the  cost  of  6  yds.  to 
the  cost  of  4  yds.  ? 

10.  9  books  will  cost  how  many  times  as  much 
as  3  books  ? 

11.  The  cost  of  3  marbles  equals  what  part  of 
the  cost  of  6  marbles  ?     Of  9  marbles  ? 

12.  The  number  of  cents  that  12  roses  cost  equals 
how  many  times  the  number  that  4  will  cost  ? 

13.  There  are  8  pts.  in  4  qts.;  how  many  8-pts. 
in  12  qts.  ? 


144  PRIMARY    ARITHMETIC. 

14.  The  number  of  ft.  in  12  yds.  equals  how 
many  halves  of  the  number  in  8  yds.  ? 

15.  A  string  15  ft.  long  is  how  many  times  as 
long  as  a  string  5  ft.  long  ? 

16.  A  string  15  ft.  long  is  how  many  times  as 
long  as'half  of  a  string  10  ft.  long  ? 

2  43  5 
2435 
2  A  §  _5 
6  12  9  15 

Practise  until  the  combinations  can  be  readily  written 
from  memory.  Try  to  secure,  in  each  combination,  the 
mental  seeing  of  the  three  figures  and  their  sum. 

This  mental  habit  greatly  lessens  the  labor  of  learning 
tables. 

What  three  equal  units  in  6  ?  In  12  ?  In  9  ? 
In  15? 

What  is  -V2-  ?  (Read  :  What  is  £  of  12  ?)  What 
is  |  ?  What  is  f  ?  What  is  -1/-  ? 


Image  three  of  each  with  the  sum.     Example : 
5 
5 

Image_5  ;  practise. 
15 


PRIMARY    ARITHMETIC.  145 

Ask  questions  and  write  answers  of  pupils    on  black- 
board. 

Questions.  Amwers. 

2  =  what  part  of  4  ?  Of  6  ?  2  =  |,  f  . 

3  =  what  part  of  6  ?  Of  9  ?  3  =  f  ,  f  . 

4  =  what  part  of  8  ?  Of  12  ?  4  =  f  ,  -1/-. 

5  =  what  part  of  10  ?  Of  15  ?  5  =  -V0-,  J^. 

Questions. 

2  =  how  many  times  1  ?  What  part  of  3  ? 

4  =  how  many  times  2  ?  What  part  of  3  ? 

6  =  how  many  times  3  ?  What  part  of  9  ? 
8  =  how  many  times  4  ?  What  part  of  12  ? 

10  =  how  many  times  5  ?       What  part  of  15  ? 


Answers. 

2  =  2  times  1,  ^a. 
4  =  2  times  2,  -2g6-. 
6  =  2  times  3,  2^. 
8  =  2  times  4, 
10  =  2  times  5, 


Questions. 

3  =  how  many  times  1  ?  How  many  halves  of  2  ? 

6  =  how  many  times  2  ?  How  many  halves  of  4  ? 

9  =  how  many  times  3  ?  How  many  halves  of  6  ? 

12  =  how  many  times  4  ?  How  many  halves  of  8  ? 

15  =  how  many  times  5  ?  How  many  halves  of  10  ? 


146 


PKIMAEY    ARITHMETIC. 


Answers. 


3  =  3  times  1, 

6  =  3  times  2,  * 

9  =  3  times  3,  ^.. 

12  =  3  times  4,  s^8-. 

15  =  3  times  5,  ^p. 

Ratios  --  Draw  units  on  the  blackboard. 


(Read  :  f  of  2.) 


4 

8 

12 

16 

20 

3 

6 

9 

12 

15 

1.  Draw  units  and  write  their  names ;  1,  2,  3,  4 ; 
2,  4,  6,  8 ;  etc. 

2.  Tell  all  you  can  about  these  units. 

3.  In  each  set  of  four  units  compare  each  unit 
with  the  other  three. 


PRIMARY    ARITHMETIC.  147 

I      O   -   T^}    7j  j    "5~~" 

6  =  2  times"  3,  ^VV2-- 
9  =  3  times  3,  *£,  a^a. 
12  =  4  times  3,  2  times  6,  -4^. 


1.  If  4  tops  cost  20/,  what  part  of  20/  will  2 
tops  cost  ?     One  top  ?     3  tops  ? 

2.  3  hats  cost  $12  ;  what  is  the  cost  of  1  hat  ? 
Of  2  hats  ?     Of  4  hats  ? 

3.  2  doz.  buttons  cost  3  dimes;  what  is  the  cost 
of  4  doz.  ?     Of  1  doz.  ?     Of  3  doz.  ? 

4.  12  Ibs.  of  butter  cost  $2  ;   what  part  of  $2 
will  3  Ibs.  cost?     What  will  6  Ibs.  cost?     What 
part  of  $2  will  9  Ibs.  cost  ? 

5.  Call  6  |  ;  what  is  12  ?     What  is  3  ?     What 
is  9? 

6.  9  boxes  of  strawberries  cost  T5/;  what  part 
of  75y  do  6  boxes  cost  ?     3  boxes  ?     12  boxes  ? 

7.  16  color  boxes  cost  a  certain  sum  ;  what  part 
of  the  sum  will  4  cost  ?     8  ?     12  ? 

8.  What  is  the  relation  of  4  to  12  ?    Of  8  to  12  ? 
Of  16  to  12  ? 

9.  A  doz.  cost  a  dime  ;  what  is  the  cost  of  4  ? 
Of  8  ?     Of  16  ? 

10.  There  are  20  things  in  a  score  ;    5  equals 
what  part  of  a  score?       10  equals  what  part? 
15  equals  what  part  ?     In  5  score  there  are  how 
many  20's  ? 

11.  5  is  -J-  of  what  unit  ?     10  equals  what  part 


148  PRIMARY    ARITHMETIC. 

of    the  unit  ?     15  equals  how  many  4ths  of  the 
unit? 


Ratios.  —  (d) 

2 

4 

3 

5 

2 

4 

3 

5 

2 

4 

3 

5 

2 

4 

3 

5 

8 

16 

12 

20 

1.  Learn  (d)  as   the  other   tables   have   been 
learned. 

2.  Compare  each  unit  with  the   other  three ; 
thus :  2  =  |,  |,  f . 

4  =  2'2,  -V-  (read,  f  of  6),  f . 
6  =  3'2,  -y-,  ¥• 
8  =  4'2,  2'4,  -V- 

3.  What  four  equal  units  in  12  ?    In  8  ?    In  16  ? 
In  20? 

4.  What  is  f  ?    -y-?    -V6-?    -Y-? 

5.  What  is  |?     -y-?     -V-?    -Y-? 

2435 

6.  Image  four  of  each  of  the  above  figures, 

5 
5 

with  the  sum.     Ex. :  Image  5 ;  practise. 

_5 
20 


PRIMARY    ARITHMETIC.  149 

Ask  questions,  and  write  pupils'  answers  on  the  black- 
board. 

Questions. 

2  =  what  part  of  4  ?  Of  6  ?  Of  8  ? 

3  =  what  part  of  6  ?  Of  9  ?  Of  12  ? 

4  =  what  part  of  8  ?  Of  12  ?  Of  16  ? 

5  =  what  part  of  10  ?  Of  15  ?  Of  20  ? 

Answers. 

—   468 
-  2?   3?  ¥' 

3  =  I,  I,  -¥-• 

4  =  f,-VW6- 

5  =  -¥->  ¥-,¥-• 

Questions. 

2  =  how  many  times  1  ?  What  part  of  3  ?     Of  4  ? 
4  =  how  many  times  2  ?  What  part  of  6  ?     Of  8  ? 
6  =  how  many  times  3  ?  What  part  of  9  ?     Of  12  ? 
8  =  how  many  times  4  ?  What  part  of  12  ?  Of  16  ? 
10  =  how  many  times  5  ?  What  part  of  15  ?  Of  20  ? 

Answers. 

2  =  2  times  1,  a^a,  f . 
4  =  2  times  2,  ^,  f . 
6  =  2  times  3,  ^-,  *£. 
8  =  2  times  4,  *^-a,  \«-. 
10  =  2  times  5,  i^-&,  -^ 

1.  What  is  V-?    Of8?    Of4? 

2.  What  is  |?     Of  12?     Of  6  ? 


150  PRIMARY    ARITHMETIC. 

3.  What  are  ^  ?     Of  9  ?     Of  6  ? 

4.  5  equals  what  part  of  10  ?     Of  20  ?     Of  15  ? 

5.  What  is  the  relation  of  10  to  20?     Of  *£ 
to  -Y-  ?     Of  -V-  to  -V-  ? 

6.  2'3's  equal  what  part  of  9  ?     Of  12  ? 

7.  6  equals  f  of  what  ?     6  equals  f  of  what  ? 

8.  16  equals  how  many  times  8  ?     How  many 
times  f  ? 

9.  What  is  the  relation  of  12  to  f  ? 

Problems.  —  1.    A  boat  sails  4  miles  in  J  hr. ; 
how  far  does  it  sail  in  1  hr.  ?     In  1J  hrs.  ? 

2.  James  is  5  yrs.  old.     His  age  equals  ^  of  his 
brother's  ;  how  old  is  his  brother  ? 

3.  $10  is  f  of  my  money,  what  is  ^  ? 

4.  If  1  apple  costs  3/?  how  many  apples  can  be 
bought  for  9/  ?     For  12/  ? 

5.  3  bonnets  cost  $9,  how  many  bonnets  can 
be  bought  for  $6  ? 

6.  If  a  family  uses  12  loaves  of  bread  in  1  week, 
what  part  of  a  week  will  9  loaves  last  ? 

7.  I  paid  ^  of  my  money  for  coal  and  the  rest 
for  flour ;  what  part  of  my  money  did  I  pay  for 
the  flour  ? 

8.  If  2  horses  eat  a  bushel  of  oats  in  a  day, 
how  much  do  3  horses  eat  ? 

9.  If  3  girls  sweep  the  floor  in  10  min.,  what 
part  of  the  floor  will  2  girls  sweep  in  the  same 
time? 


PRIMARY    ARITHMETIC.  151 

10.  3  oranges  cost  5/,  how  many  *5^  will  12 
oranges  cost  ?     A  doz.  ? 

11.  John  has  6/,  and  his  brother  has  2  times 
as  many  ;  how  many  has  his  brother  ?      The  two 
boys  have  how  many  6/  ? 

12.  3  pairs  of  shoes  cost  $9,  how  many  $9  will 
6  pairs  cost  ? 

13.  If  it  takes  15  boys  one  day  to  dig  a  ditch, 
what  part  of  the  ditch  can  5  boys  dig  in  1  day  ? 
How  many  days  will  it  take  the  5  boys  to  dig  the 
other  f  of  the  ditch  ? 

14.  At  $  J  a  bushel,  how  many  bushels  of  apples 
can  be  bought  for  $3  ? 

15.  Mary  is  5  yrs.  old.     Jane  is  4  times  as  old, 
How  old  is  Jane  ? 

16.  Roy  walks  2  blocks,  while  his  sister  walks 

1  block.     Roy  walks  how  many  times  as  fast  as 
his  sister  ?      If  Roy  can  walk  to  school  in  4  min.? 
it  will  take  his  sister  how  many  4-min.  ?      How 
many  min.?     If  John  walks  3  times  as  fast  as  Roy, 
John  will  walk  how  far,  while  Roy  is  walking 

2  blocks  ? 

17.  Draw  a  line  and  call  it  the  distance  Roy 
walks  in  1  min.     Draw  another,  showing  how  far 
his  sister  walks   in   the  same  time.      Draw  one 
showing  how  far  John  walks  in  the  same  time  ? 

18.  Caroline  has  8  roses ;    to  how  many  little 
girls  can  she  give  2  and  yet  keep  2  herself  ? 

19.  Nettie  and  Addie  are  in  the  middle  of  the 


152  PRIMARY    ARITHMETIC. 

room.     If  *Addie  walks  3  yds.  north,  and  Nettie 
2  yds.  south,  how  far  apart  will  they  be  ? 

20.  A  boy  had  20/,  and  lost  16/;  what  part  of 
his  money  did  he  lose  ? 

21.  Mr.  Jones  lives  4  blocks  east  of  the  school- 
house,  and  Mr.  Brown  3  blocks  west;    how  far 
apart  do  they  live  ? 

22.  Howard  and  Frank  bought  a  box  of  marbles 
for  6/.     Howard  paid  4/,  and  Frank  2/ ;    what 
part  of  the  marbles  ought  each  to  have  ? 

23.  A  boy  sells  papers  at  2/  each;  how  many 
does  he  sell  to  receive  10/  ?     To  receive  8/  ? 

24.  If  he  sells  £  of  10  papers,  he  will  receive 
how  many  cents  ?     If  he  sells  f  of  10  papers  ? 

25.  A  lady  gave   to   Carrie   6  apples  and   to 
Fannie  f  of  8  apples ;  to  which  did  she  give  the 
greater  number  ? 

26.  4  peaches  equal  what  part  of  6  peaches  ? 
Of  8  peaches  ?     Of  10  peaches  ? 

27.  The  cost  of  6  peaches  equals  what  part  of 
the  cost  of  8  ?     Of  10  ? 

28.  The  jelly  that  4  peaches  will  make  equals 
what  part  of  the  jelly  that  6  peaches  will  make  ? 
That  10  will  make  ?     That  8  will  make  ? 

29.  How  many  pts.  equal  1  qt.  ?  A  gallon  equals 
how  many  qts.  ?     A  year  equals  how  many  6-mos.  ? 
How  many  4-mos.  ?     How  many  3-mos.  ?     How 
many  6's  in  a  dozen  ?      How  many  4's  ?     How 
many  3's  ? 


PRIMARY    ARITHMETIC.  153 

30.  A  dime  equals  how  many  nickels  ?    2  dimes 
equal  how  many  nickels  ? 

31.  1  yd.  equals  how  many  ft.  ?     5  yds.  equal 
how  many  3-ft.  ?     How  many  yds.  in  6  ft.  ?     In 
9  ft.  ?     In  12  ft.  ? 

32.  If  a  yd.  of  cord  is  worth  15/,  what  are  3  ft. 
of  cord  worth  ? 

33.  In  a  score  there  are  how  many  10's  ?    How 
many  5's  ?      10  equals  what  part  of  a  score  ?      15 
equals  what  part  of  a  score  ? 

34.  There  are  7  days  in  a  week  ;   how  many 
7-day s  in   9  weeks  ?      14  days  equal  how  many 
weeks  ?     7  days  equal  what  part  of  3  weeks  ? 

35.  There  are  5  school  days  in  a  week;   how 
many  school  days  in  3  weeks  ? 

36.  9  tons  of  coal  last  a  family  6  mos. ;   how 
many  9-tons  will  last  a  yr.  ?     How  many  tons  ? 

37.  If  2  barrels  of  flour  last  4  mos.,  how  many 
2-barrels  will  last  a  yr.  ?     How  many  barrels  ? 

38.  If   you   pour   a   pint   of   milk   into   a  qt. 
measure,  it  fills  what  part  of  it  ? 

39.  Mr.  Robinson  sells  2  pts.  of  milk  for  6/; 
how  much  ought  he  to  receive  for  a  qt.  ? 

40.  2  qts.  of  water  fill  what  part  of  a  gallon 
measure?      3  qts.  of  water  fill  what  part  of  a 
gallon  measure  ? 

41.  If  you  take  a  qt.  of  milk  out  of  a  gallon  of 
milk,  what  part  of  a  gallon  remains  ? 

42.  From   a   piece    of    cloth    20   yds.    long   a 


154  PRIMARY   ARITHMETIC. 

merchant  cuts  5  yds. ;  the  smaller  piece  equals 
what  part  of  the  larger  piece  ?  How  many  5-yards 
in  the  larger  piece  ?  How  many  yards  ? 

43.  A  lady  paid  $8  for  a  dress.     She  paid  $4 
more  for  a  cloak  than  for  the  dress ;  how  much 
did  she  pay  for  the  cloak  ?     How  much  for  both  ? 

44.  There  are  4  gills  in  1  pt.     How  many  gills 
in  3  pts.  ?     In  £  pt.  ?     In  4  pts.  ?     2  gills  equal 
what  part  of  a  pint  ?    How  many  pints  in  8  gills  ? 
In  12  ?     In  6  ?     6  gills  equal  how  many  halves  of 
a  pint  ?    In  a  qt.  there  are  how  many  gills  ?    A  qt, 
is  how  many  times  as  much  as  a  gill  ? 

45.  How  many  6-in.  in  1|-  ft.  ?      How  many 
inches  ? 

46.  How  many  half-dozen  in  1  doz.  and  6  ? 

47.  What  is  the  sum  of  a  dime  and  4/?     Of 
2  dimes  and  a  5/  piece  ? 

48.  Two  dimes  equal  how  many  nickels  ? 

49.  The  candy  that  a  nickel  will  buy  equals 
what  part  of  the  candy  that  can  be  bought  for 
2  dimes  ? 


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